{"id":14608,"date":"2018-09-27T18:14:43","date_gmt":"2018-09-27T18:14:43","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/precalculus\/chapter\/solving-systems-with-cramers-rule\/"},"modified":"2025-02-05T05:19:53","modified_gmt":"2025-02-05T05:19:53","slug":"solving-systems-with-cramers-rule","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/precalculus\/chapter\/solving-systems-with-cramers-rule\/","title":{"raw":"Solving Systems with Cramer's Rule","rendered":"Solving Systems with Cramer&#8217;s Rule"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\nBy the end of this section, you will be able to:\r\n<ul>\r\n \t<li>Evaluate 2 \u00d7 2 determinants.<\/li>\r\n \t<li>Use Cramer\u2019s Rule to solve a system of equations in two variables.<\/li>\r\n \t<li>Evaluate 3 \u00d7 3 determinants.<\/li>\r\n \t<li>Use Cramer\u2019s Rule to solve a system of three equations in three variables.<\/li>\r\n \t<li>Know the properties of determinants.<\/li>\r\n<\/ul>\r\n<\/div>\r\nWe have learned how to solve systems of equations in two variables and three variables, and by multiple methods: substitution, addition, Gaussian elimination, using the inverse of a matrix, and graphing. Some of these methods are easier to apply than others and are more appropriate in certain situations. In this section, we will study two more strategies for solving systems of equations.\r\n<h2>Using Cramer\u2019s Rule to Solve a System of Two Equations in Two Variables<\/h2>\r\n<h2>Evaluating the Determinant of a 2\u00d72 Matrix<\/h2>\r\nA determinant is a real number that can be very useful in mathematics because it has multiple applications, such as calculating area, volume, and other quantities. Here, we will use determinants to reveal whether a matrix is invertible by using the entries of a <strong>square matrix<\/strong> to determine whether there is a solution to the system of equations. Perhaps one of the more interesting applications, however, is their use in cryptography. Secure signals or messages are sometimes sent encoded in a matrix. The data can only be decrypted with an <strong>invertible matrix<\/strong> and the determinant. For our purposes, we focus on the determinant as an indication of the invertibility of the matrix. Calculating the determinant of a matrix involves following the specific patterns that are outlined in this section.\r\n<div class=\"textbox shaded\">\r\n<h3>A General Note: Find the Determinant of a 2 \u00d7 2 Matrix<\/h3>\r\nThe <strong>determinant<\/strong> of a [latex]2\\text{ }\\times \\text{ }2[\/latex] matrix, given\r\n<div style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{cc}a&amp; b\\\\ c&amp; d\\end{array}\\right][\/latex]<\/div>\r\nis defined as\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181431\/CNX_Precalc_Figure_09_08_0012.jpg\" alt=\"\" width=\"487\" height=\"59\" \/> <b>Figure 1<\/b>[\/caption]\r\n\r\nNotice the change in notation. There are several ways to indicate the determinant, including [latex]\\mathrm{det}\\left(A\\right)[\/latex] and replacing the brackets in a matrix with straight lines, [latex]|A|[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 1: Finding the Determinant of a 2 \u00d7 2 Matrix<\/h3>\r\nFind the determinant of the given matrix.\r\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{cc}5&amp; 2\\\\ -6&amp; 3\\end{array}\\right][\/latex]<\/p>\r\n[reveal-answer q=\"149250\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"149250\"]\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\mathrm{det}\\left(A\\right)&amp;=\\left\\rvert\\begin{array}{cc}5&amp; 2\\\\ -6&amp; 3\\end{array}\\right\\rvert\\hfill \\\\ &amp;=5\\left(3\\right)-\\left(-6\\right)\\left(2\\right)\\hfill \\\\ &amp;=27\\hfill \\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]6397[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Using Cramer\u2019s Rule to Solve a System of Two Equations in Two Variables<\/h2>\r\nWe will now introduce a final method for solving systems of equations that uses determinants. Known as <strong>Cramer\u2019s Rule<\/strong>, this technique dates back to the middle of the 18th century and is named for its innovator, the Swiss mathematician Gabriel Cramer (1704\u20131752), who introduced it in 1750 in Introduction \u00e0 l'Analyse des lignes Courbes alg\u00e9briques. Cramer\u2019s Rule is a viable and efficient method for finding solutions to systems with an arbitrary number of unknowns, provided that we have the same number of equations as unknowns.\r\n\r\nCramer\u2019s Rule will give us the unique solution to a system of equations, if it exists. However, if the system has no solution or an infinite number of solutions, this will be indicated by a determinant of zero. To find out if the system is inconsistent or dependent, another method, such as elimination, will have to be used.\r\n\r\nTo understand Cramer\u2019s Rule, let\u2019s look closely at how we solve systems of linear equations using basic row operations. Consider a system of two equations in two variables.\r\n<div style=\"text-align: center;\">[latex]\\begin{align}{a}_{1}x+{b}_{1}y&amp;={c}_{1}&amp;&amp;R_{1}\\\\ {a}_{2}x+{b}_{2}y&amp;={c}_{2}&amp;&amp;R_2\\end{align}[\/latex]<\/div>\r\nWe eliminate one variable using row operations and solve for the other. Say that we wish to solve for [latex]x[\/latex]. If equation (2) is multiplied by the opposite of the coefficient of [latex]y[\/latex] in equation (1), equation (1) is multiplied by the coefficient of [latex]y[\/latex] in equation (2), and we add the two equations, the variable [latex]y[\/latex] will be eliminated.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}b_{2}a_{1}x+b_{2}b_{1}y&amp;=b_{2}c_{1} \\\\ \u2212b_{1}a_{2}x\u2212b_{1}b_{2}y&amp;=\u2212b_{1}c_{2} \\\\ \\hline b_{2}a_{1}x\u2212b_{1}a_{2}x&amp;=\u2212b_{2}c_{1}\u2212b_{1}c_{2}\\end{align}[\/latex] [latex]\\begin{align}&amp;\\text{Multiply }R_{1}\\text{ by }b_{2} \\\\ &amp;\\text{Multiply }R_{2}\\text{ by }\u2212b_{2} \\\\ \\text{ } \\end{align}[\/latex]<\/p>\r\nNow, solve for [latex]x[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{gathered}{b}_{2}{a}_{1}x-{b}_{1}{a}_{2}x={b}_{2}{c}_{1}-{b}_{1}{c}_{2}\\\\ \\hfill \\\\ x\\left({b}_{2}{a}_{1}-{b}_{1}{a}_{2}\\right)={b}_{2}{c}_{1}-{b}_{1}{c}_{2}\\\\ \\hfill \\\\ x=\\frac{{b}_{2}{c}_{1}-{b}_{1}{c}_{2}}{{b}_{2}{a}_{1}-{b}_{1}{a}_{2}}=\\frac{\\left\\rvert\\begin{array}{cc}{c}_{1}&amp; {b}_{1}\\\\ {c}_{2}&amp; {b}_{2}\\end{array}\\right\\rvert}{\\left\\rvert\\begin{array}{cc}{a}_{1}&amp; {b}_{1}\\\\ {a}_{2}&amp; {b}_{2}\\end{array}\\right\\rvert}\\hfill \\end{gathered}[\/latex]<\/div>\r\nSimilarly, to solve for [latex]y[\/latex], we will eliminate [latex]x[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{align}a_{2}a_{1}x+a_{2}b_{1}y&amp;=a_{2}c_{1} \\\\\u2212a_{1}a_{2}x\u2212a_{1}b_{2}y&amp;=\u2212a_{1}c_{2} \\\\ \\hline a_{2}b_{1}y\u2212a_{1}b_{2}y&amp;=a_{2}c_{1}\u2212a_{1}c_{2}\\end{align}[\/latex] [latex]\\begin{align}&amp;\\text{Multiply }R_{1}\\text{ by }a_{2} \\\\&amp;\\text{Multiply }R_{2}\\text{ by }\u2212a_{1} \\\\ \\text{ } \\end{align}[\/latex]<\/div>\r\nSolving for [latex]y[\/latex] gives\r\n<div style=\"text-align: center;\">[latex]\\begin{gathered}{a}_{2}{b}_{1}y-{a}_{1}{b}_{2}y={a}_{2}{c}_{1}-{a}_{1}{c}_{2} \\\\ y\\left({a}_{2}{b}_{1}-{a}_{1}{b}_{2}\\right)={a}_{2}{c}_{1}-{a}_{1}{c}_{2} \\\\ y=\\frac{{a}_{2}{c}_{1}-{a}_{1}{c}_{2}}{{a}_{2}{b}_{1}-{a}_{1}{b}_{2}}=\\frac{{a}_{1}{c}_{2}-{a}_{2}{c}_{1}}{{a}_{1}{b}_{2}-{a}_{2}{b}_{1}}=\\frac{\\left\\rvert\\begin{array}{cc}{a}_{1}&amp; {c}_{1}\\\\ {a}_{2}&amp; {c}_{2}\\end{array}\\right\\rvert}{\\left\\rvert\\begin{array}{cc}{a}_{1}&amp; {b}_{1}\\\\ {a}_{2}&amp; {b}_{2}\\end{array}\\right\\rvert} \\end{gathered}[\/latex]<\/div>\r\nNotice that the denominator for both [latex]x[\/latex] and [latex]y[\/latex] is the determinant of the coefficient matrix.\r\n\r\nWe can use these formulas to solve for [latex]x[\/latex] and [latex]y[\/latex], but Cramer\u2019s Rule also introduces new notation:\r\n<ul>\r\n \t<li>[latex]D:[\/latex] determinant of the coefficient matrix<\/li>\r\n \t<li>[latex]{D}_{x}:[\/latex] determinant of the numerator in the solution of [latex]x[\/latex]\r\n<div style=\"text-align: center;\">[latex]x=\\frac{{D}_{x}}{D}[\/latex]<\/div><\/li>\r\n \t<li>[latex]{D}_{y}:[\/latex] determinant of the numerator in the solution of [latex]y[\/latex]\r\n<div style=\"text-align: center;\">[latex]y=\\frac{{D}_{y}}{D}[\/latex]<\/div><\/li>\r\n<\/ul>\r\nThe key to Cramer\u2019s Rule is replacing the variable column of interest with the constant column and calculating the determinants. We can then express [latex]x[\/latex] and [latex]y[\/latex] as a quotient of two determinants.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Cramer\u2019s Rule for 2\u00d72 Systems<\/h3>\r\n<strong>Cramer\u2019s Rule<\/strong> is a method that uses determinants to solve systems of equations that have the same number of equations as variables.\r\n\r\nConsider a system of two linear equations in two variables.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}{a}_{1}x+{b}_{1}y={c}_{1}\\\\ {a}_{2}x+{b}_{2}y={c}_{2}\\end{array}[\/latex]<\/p>\r\nThe solution using Cramer\u2019s Rule is given as\r\n<p style=\"text-align: center;\">[latex]x=\\frac{{D}_{x}}{D}=\\frac{\\left\\rvert\\begin{array}{cc}{c}_{1}&amp; {b}_{1}\\\\ {c}_{2}&amp; {b}_{2}\\end{array}\\right\\rvert}{\\left\\rvert\\begin{array}{cc}{a}_{1}&amp; {b}_{1}\\\\ {a}_{2}&amp; {b}_{2}\\end{array}\\right\\rvert},D\\ne 0;\\text{ }\\text{ }y=\\frac{{D}_{y}}{D}=\\frac{\\left\\rvert\\begin{array}{cc}{a}_{1}&amp; {c}_{1}\\\\ {a}_{2}&amp; {c}_{2}\\end{array}\\right\\rvert}{\\left\\rvert\\begin{array}{cc}{a}_{1}&amp; {b}_{1}\\\\ {a}_{2}&amp; {b}_{2}\\end{array}\\right\\rvert},D\\ne 0[\/latex].<\/p>\r\nIf we are solving for [latex]x[\/latex], the [latex]x[\/latex] column is replaced with the constant column. If we are solving for [latex]y[\/latex], the [latex]y[\/latex] column is replaced with the constant column.\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 2: Using Cramer\u2019s Rule to Solve a 2 \u00d7 2 System<\/h3>\r\nSolve the following [latex]2\\text{ }\\times \\text{ }2[\/latex] system using Cramer\u2019s Rule.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}12x+3y&amp;=15\\\\ 2x - 3y&amp;=13\\end{align}[\/latex]<\/p>\r\n[reveal-answer q=\"472105\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"472105\"]\r\n\r\nSolve for [latex]x[\/latex].\r\n<p style=\"text-align: center;\">[latex]x=\\frac{{D}_{x}}{D}=\\frac{\\left\\rvert\\begin{array}{rr}\\hfill 15&amp; \\hfill 3\\\\ \\hfill 13&amp; \\hfill -3\\end{array}\\right\\rvert}{\\left\\rvert\\begin{array}{rr}\\hfill 12&amp; \\hfill 3\\\\ \\hfill 2&amp; \\hfill -3\\end{array}\\right\\rvert}=\\frac{-45 - 39}{-36 - 6}=\\frac{-84}{-42}=2[\/latex]<\/p>\r\nSolve for [latex]y[\/latex].\r\n<p style=\"text-align: center;\">[latex]y=\\frac{{D}_{y}}{D}=\\frac{\\left\\rvert\\begin{array}{rr}\\hfill 12&amp; \\hfill 15\\\\ \\hfill 2&amp; \\hfill 13\\end{array}\\right\\rvert}{\\left\\rvert\\begin{array}{rr}\\hfill 12&amp; \\hfill 3\\\\ \\hfill 2&amp; \\hfill -3\\end{array}\\right\\rvert}=\\frac{156 - 30}{-36 - 6}=-\\frac{126}{42}=-3[\/latex]<\/p>\r\nThe solution is [latex]\\left(2,-3\\right)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nUse Cramer\u2019s Rule to solve the 2 \u00d7 2 system of equations.\r\n<p style=\"text-align: center;\">[latex]\\begin{gathered}x+2y=-11 \\\\ -2x+y=-13 \\end{gathered}[\/latex]<\/p>\r\n[reveal-answer q=\"498348\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"498348\"]\r\n\r\n[latex]\\left(3,-7\\right)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]174705[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Using Cramer\u2019s Rule to Solve a System of Three Equations in Three Variables<\/h2>\r\n<h2>Evaluating the Determinant of a 3 \u00d7 3 Matrix<\/h2>\r\nFinding the determinant of a 2\u00d72 matrix is straightforward, but finding the determinant of a 3\u00d73 matrix is more complicated. One method is to augment the 3\u00d73 matrix with a repetition of the first two columns, giving a 3\u00d75 matrix. Then we calculate the sum of the products of entries <em>down<\/em> each of the three diagonals (upper left to lower right), and subtract the products of entries <em>up<\/em> each of the three diagonals (lower left to upper right). This is more easily understood with a visual and an example.\r\n\r\nFind the <strong>determinant<\/strong> of the 3\u00d73 matrix.\r\n<div style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{ccc}{a}_{1}&amp; {b}_{1}&amp; {c}_{1}\\\\ {a}_{2}&amp; {b}_{2}&amp; {c}_{2}\\\\ {a}_{3}&amp; {b}_{3}&amp; {c}_{3}\\end{array}\\right][\/latex]<\/div>\r\n<ol>\r\n \t<li>Augment [latex]A[\/latex] with the first two columns.\r\n<div style=\"text-align: center;\">[latex]\\mathrm{det}\\left(A\\right)=\\left\\rvert\\begin{array}{ccc}{a}_{1}&amp; {b}_{1}&amp; {c}_{1}\\\\ {a}_{2}&amp; {b}_{2}&amp; {c}_{2}\\\\ {a}_{3}&amp; {b}_{3}&amp; {c}_{3}\\end{array}\\right\\rvert \\left.\\begin{array}{c}{a}_{1}\\\\ {a}_{2}\\\\ {a}_{3}\\end{array}\\begin{array}{c}{b}_{1}\\\\ {b}_{2}\\\\ {b}_{3}\\end{array}\\right\\rvert[\/latex]<\/div><\/li>\r\n \t<li>From upper left to lower right: Multiply the entries down the first diagonal. Add the result to the product of entries down the second diagonal. Add this result to the product of the entries down the third diagonal.<\/li>\r\n \t<li>From lower left to upper right: Subtract the product of entries up the first diagonal. From this result subtract the product of entries up the second diagonal. From this result, subtract the product of entries up the third diagonal.<\/li>\r\n<\/ol>\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181433\/CNX_Precalc_Figure_09_08_0022.jpg\" alt=\"\" width=\"487\" height=\"89\" \/> <b>Figure 2<\/b>[\/caption]\r\n\r\nThe algebra is as follows:\r\n<div style=\"text-align: center;\">[latex]|A|={a}_{1}{b}_{2}{c}_{3}+{b}_{1}{c}_{2}{a}_{3}+{c}_{1}{a}_{2}{b}_{3}-{a}_{3}{b}_{2}{c}_{1}-{b}_{3}{c}_{2}{a}_{1}-{c}_{3}{a}_{2}{b}_{1}[\/latex]<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 3: Finding the Determinant of a 3 \u00d7 3 Matrix<\/h3>\r\nFind the determinant of the 3 \u00d7 3 matrix given\r\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{ccc}0&amp; 2&amp; 1\\\\ 3&amp; -1&amp; 1\\\\ 4&amp; 0&amp; 1\\end{array}\\right][\/latex]<\/p>\r\n[reveal-answer q=\"915069\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"915069\"]\r\n\r\nAugment the matrix with the first two columns and then follow the formula. Thus,\r\n<p style=\"text-align: center;\">[latex]\\begin{align}|A|&amp;=\\left\\rvert\\begin{array}{ccc}0&amp; 2&amp; 1\\\\ 3&amp; -1&amp; 1\\\\ 4&amp; 0&amp; 1\\end{array}\\right\\rvert\\left.\\begin{array}{c}0 &amp; 2\\\\ 3 &amp; -1\\\\ 4 &amp; 0\\end{array}\\right\\rvert\\hfill \\\\ &amp;=0\\left(-1\\right)\\left(1\\right)+2\\left(1\\right)\\left(4\\right)+1\\left(3\\right)\\left(0\\right)-4\\left(-1\\right)\\left(1\\right)-0\\left(1\\right)\\left(0\\right)-1\\left(3\\right)\\left(2\\right)\\hfill \\\\ &amp;=0+8+0+4 - 0-6\\hfill \\\\ &amp;=6\\hfill \\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nFind the determinant of the 3 \u00d7 3 matrix.\r\n<p style=\"text-align: center;\">[latex]\\mathrm{det}\\left(A\\right)=\\left\\rvert\\begin{array}{ccc}1&amp; -3&amp; 7\\\\ 1&amp; 1&amp; 1\\\\ 1&amp; -2&amp; 3\\end{array}\\right\\rvert[\/latex]<\/p>\r\n[reveal-answer q=\"612489\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"612489\"]\r\n\r\n[latex]-10[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]19398[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Q &amp; A<\/h3>\r\n<h3>Can we use the same method to find the determinant of a larger matrix?<\/h3>\r\n<em>Yes, but for larger matrices it is best to use a graphing utility or computer software.<\/em>\r\n\r\n<\/div>\r\n<h2>Using Cramer\u2019s Rule to Solve a System of Three Equations in Three Variables<\/h2>\r\nNow that we can find the <strong>determinant<\/strong> of a 3 \u00d7 3 matrix, we can apply <strong>Cramer\u2019s Rule<\/strong> to solve a <strong>system of three equations in three variables<\/strong>. Cramer\u2019s Rule is straightforward, following a pattern consistent with Cramer\u2019s Rule for 2 \u00d7 2 matrices. As the order of the matrix increases to 3 \u00d7 3, however, there are many more calculations required.\r\n\r\nWhen we calculate the determinant to be zero, Cramer\u2019s Rule gives no indication as to whether the system has no solution or an infinite number of solutions. To find out, we have to perform elimination on the system.\r\n\r\nConsider a 3 \u00d7 3 system of equations.\r\n\r\n[caption id=\"attachment_12665\" align=\"aligncenter\" width=\"243\"]<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/145\/2017\/01\/06215037\/Screen-Shot-2017-01-06-at-1.50.12-PM.png\"><img class=\"size-full wp-image-12665\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181435\/Screen-Shot-2017-01-06-at-1.50.12-PM.png\" alt=\"\" width=\"243\" height=\"136\" \/><\/a> <strong>Figure 3<\/strong>[\/caption]\r\n\r\n<div style=\"text-align: center;\">[latex]x=\\frac{{D}_{x}}{D},y=\\frac{{D}_{y}}{D},z=\\frac{{D}_{z}}{D},D\\ne 0[\/latex]<\/div>\r\nwhere\r\n\r\n[caption id=\"attachment_12667\" align=\"aligncenter\" width=\"856\"]<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/145\/2017\/01\/06215135\/Screen-Shot-2017-01-06-at-1.51.06-PM.png\"><img class=\"size-full wp-image-12667\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181437\/Screen-Shot-2017-01-06-at-1.51.06-PM.png\" alt=\"\" width=\"856\" height=\"128\" \/><\/a> <strong>Figure 4<\/strong>[\/caption]\r\n\r\nIf we are writing the determinant [latex]{D}_{x}[\/latex], we replace the [latex]x[\/latex] column with the constant column. If we are writing the determinant [latex]{D}_{y}[\/latex], we replace the [latex]y[\/latex] column with the constant column. If we are writing the determinant [latex]{D}_{z}[\/latex], we replace the [latex]z[\/latex] column with the constant column. Always check the answer.\r\n<div class=\"textbox shaded\">\r\n<h3>Example 4: Solving a 3 \u00d7 3 System Using Cramer\u2019s Rule<\/h3>\r\nFind the solution to the given 3 \u00d7 3 system using Cramer\u2019s Rule.\r\n<p style=\"text-align: center;\">[latex]\\begin{gathered}x+y-z=6\\\\ 3x - 2y+z=-5\\\\ x+3y - 2z=14\\end{gathered}[\/latex]<\/p>\r\n[reveal-answer q=\"180631\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"180631\"]\r\n\r\nUse Cramer\u2019s Rule.\r\n<p style=\"text-align: center;\">[latex]D=\\left\\rvert\\begin{array}{ccc}1&amp; 1&amp; -1\\\\ 3&amp; -2&amp; 1\\\\ 1&amp; 3&amp; -2\\end{array}\\right\\rvert\\text{, }{D}_{x}=\\left\\rvert\\begin{array}{ccc}6&amp; 1&amp; -1\\\\ -5&amp; -2&amp; 1\\\\ 14&amp; 3&amp; -2\\end{array}\\right\\rvert\\text{, }{D}_{y}=\\left\\rvert\\begin{array}{ccc}1&amp; 6&amp; -1\\\\ 3&amp; -5&amp; 1\\\\ 1&amp; 14&amp; -2\\end{array}\\right\\rvert\\text{, }{D}_{z}=\\left\\rvert\\begin{array}{ccc}1&amp; 1&amp; 6\\\\ 3&amp; -2&amp; -5\\\\ 1&amp; 3&amp; 14\\end{array}\\right\\rvert[\/latex]<\/p>\r\nThen,\r\n<p style=\"text-align: center;\">[latex]\\begin{align}x&amp;=\\frac{{D}_{x}}{D}=\\frac{-3}{-3}=1\\hfill \\\\ y&amp;=\\frac{{D}_{y}}{D}=\\frac{-9}{-3}=3\\hfill \\\\ z&amp;=\\frac{{D}_{z}}{D}=\\frac{6}{-3}=-2\\hfill \\end{align}[\/latex]<\/p>\r\nThe solution is [latex]\\left(1,3,-2\\right)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nUse Cramer\u2019s Rule to solve the 3 \u00d7 3 matrix.\r\n<p style=\"text-align: center;\">[latex]\\begin{gathered} x - 3y+7z=13\\\\ x+y+z=1\\\\ x - 2y+3z=4\\end{gathered}[\/latex]<\/p>\r\n[reveal-answer q=\"700599\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"700599\"]\r\n\r\n[latex]\\left(-2,\\frac{3}{5},\\frac{12}{5}\\right)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 5: Using Cramer\u2019s Rule to Solve an Inconsistent System<\/h3>\r\nSolve the system of equations using Cramer\u2019s Rule.\r\n<p style=\"text-align: center;\">[latex]\\begin{gathered}3x - 2y=4 \\\\ 6x - 4y=0\\end{gathered}[\/latex]<\/p>\r\n[reveal-answer q=\"966288\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"966288\"]\r\n\r\nWe begin by finding the determinants [latex]D,{D}_{x},\\text{and }{D}_{y}[\/latex].\r\n<p style=\"text-align: center;\">[latex]D=\\left\\rvert\\begin{array}{cc}3&amp; -2\\\\ 6&amp; -4\\end{array}\\right\\rvert=3\\left(-4\\right)-6\\left(-2\\right)=0[\/latex]<\/p>\r\nWe know that a determinant of zero means that either the system has no solution or it has an infinite number of solutions. To see which one, we use the process of elimination. Our goal is to eliminate one of the variables.\r\n<ol>\r\n \t<li>Multiply equation (1) by [latex]-2[\/latex].<\/li>\r\n \t<li>Add the result to equation [latex]\\left(2\\right)[\/latex].<\/li>\r\n<\/ol>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\u22126x+4y&amp;=\u22128 \\\\ 6x\u22124y&amp;=0 \\\\ \\hline0&amp;=-8\\end{align}[\/latex]<\/p>\r\nWe obtain the equation [latex]0=-8[\/latex], which is false. Therefore, the system has no solution. Graphing the system reveals two parallel lines.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181440\/CNX_Precalc_Figure_09_08_0032.jpg\" alt=\"Graph of two parallel lines with the equations y=three-halves x and y=three-halves x minus 2.\" width=\"487\" height=\"441\" \/> <b>Figure 5<\/b>[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 6: Use Cramer\u2019s Rule to Solve a Dependent System<\/h3>\r\nSolve the system with an infinite number of solutions.\r\n<p style=\"text-align: center;\">[latex]\\begin{gathered} x - 2y+3z=0\\\\ 3x+y - 2z=0 \\\\ 2x - 4y+6z=0 \\end{gathered}[\/latex]<\/p>\r\n[reveal-answer q=\"282282\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"282282\"]\r\n\r\nLet\u2019s find the determinant first. Set up a matrix augmented by the first two columns.\r\n<p style=\"text-align: center;\">[latex]\\left\\rvert\\begin{array}{rrr}\\hfill 1&amp; \\hfill -2&amp; \\hfill 3\\\\ \\hfill 3&amp; \\hfill 1&amp; \\hfill -2\\\\ \\hfill 2&amp; \\hfill -4&amp; \\hfill 6\\end{array}\\right\\rvert\\left.\\begin{array}{rr}\\hfill 1&amp; \\hfill -2\\\\ \\hfill 3&amp; \\hfill 1\\\\ \\hfill 2&amp; \\hfill -4\\end{array}\\right\\rvert[\/latex]<\/p>\r\nThen,\r\n<p style=\"text-align: center;\">[latex]1\\left(1\\right)\\left(6\\right)+\\left(-2\\right)\\left(-2\\right)\\left(2\\right)+3\\left(3\\right)\\left(-4\\right)-2\\left(1\\right)\\left(3\\right)-\\left(-4\\right)\\left(-2\\right)\\left(1\\right)-6\\left(3\\right)\\left(-2\\right)=0[\/latex]<\/p>\r\nAs the determinant equals zero, there is either no solution or an infinite number of solutions. We have to perform elimination to find out.\r\n<ol>\r\n \t<li>Multiply equation (1) by [latex]-2[\/latex] and add the result to equation (3):\r\n<div style=\"text-align: center;\">[latex]\\begin{align} -2x+4y - 6z&amp;=0\\\\ 2x - 4y+6z&amp;=0\\\\ \\hline 0&amp;=0 \\end{align}[\/latex]<\/div><\/li>\r\n \t<li>Obtaining an answer of [latex]0=0[\/latex], a statement that is always true, means that the system has an infinite number of solutions. Graphing the system, we can see that two of the planes are the same and they both intersect the third plane on a line.<\/li>\r\n<\/ol>\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181442\/CNX_Precalc_Figure_09_08_0052.jpg\" alt=\"Two planes intersecting a third plane. One plane's equation is x minus 2y plus 3z equals zero. The second plane's equation is 2x minus 4y plus 6z equals zero. The third plane's equation is 3x plus y plus 2z equals zero.\" width=\"487\" height=\"214\" \/> <b>Figure 6<\/b>[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]174706[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Understanding Properties of Determinants<\/h2>\r\nThere are many <strong>properties of determinants<\/strong>. Listed here are some properties that may be helpful in calculating the determinant of a matrix.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Properties of Determinants<\/h3>\r\n<ol>\r\n \t<li>If the matrix is in upper triangular form, the determinant equals the product of entries down the main diagonal.<\/li>\r\n \t<li>When two rows are interchanged, the determinant changes sign.<\/li>\r\n \t<li>If either two rows or two columns are identical, the determinant equals zero.<\/li>\r\n \t<li>If a matrix contains either a row of zeros or a column of zeros, the determinant equals zero.<\/li>\r\n \t<li>The determinant of an inverse matrix [latex]{A}^{-1}[\/latex] is the reciprocal of the determinant of the matrix [latex]A[\/latex].<\/li>\r\n \t<li>If any row or column is multiplied by a constant, the determinant is multiplied by the same factor.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 7: Illustrating Properties of Determinants<\/h3>\r\nIllustrate each of the properties of determinants.\r\n\r\n[reveal-answer q=\"123702\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"123702\"]\r\n\r\nProperty 1 states that if the matrix is in upper triangular form, the determinant is the product of the entries down the main diagonal.\r\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{rrr}\\hfill 1&amp; \\hfill 2&amp; \\hfill 3\\\\ \\hfill 0&amp; \\hfill 2&amp; \\hfill 1\\\\ \\hfill 0&amp; \\hfill 0&amp; \\hfill -1\\end{array}\\right][\/latex]<\/p>\r\nAugment [latex]A[\/latex] with the first two columns.\r\n<p style=\"text-align: center;\">[latex]A=\\left[\\left.\\begin{array}{ccc}1&amp; 2&amp; 3\\\\ 0&amp; 2&amp; 1\\\\ 0&amp; 0&amp; -1\\end{array}\\right\\rvert\\begin{array}{c}1\\\\ 0\\\\ 0\\end{array}\\begin{array}{c}2\\\\ 2\\\\ 0\\end{array}\\right][\/latex]<\/p>\r\nThen\r\n<p style=\"text-align: center;\">[latex]\\mathrm{det}\\left(A\\right)=1\\left(2\\right)\\left(-1\\right)+2\\left(1\\right)\\left(0\\right)+3\\left(0\\right)\\left(0\\right)-0\\left(2\\right)\\left(3\\right)-0\\left(1\\right)\\left(1\\right)+1\\left(0\\right)\\left(2\\right)=-2 [\/latex]<\/p>\r\nProperty 2 states that interchanging rows changes the sign. Given\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\begin{array}{l}\\\\ A=\\left[\\begin{array}{cc}-1&amp; 5\\\\ 4&amp; -3\\end{array}\\right],\\mathrm{det}\\left(A\\right)=\\left(-1\\right)\\left(-3\\right)-\\left(4\\right)\\left(5\\right)=3 - 20=-17\\end{array}\\hfill \\\\ \\hfill \\\\ B=\\left[\\begin{array}{cc}4&amp; -3\\\\ -1&amp; 5\\end{array}\\right],\\mathrm{det}\\left(B\\right)=\\left(4\\right)\\left(5\\right)-\\left(-1\\right)\\left(-3\\right)=20 - 3=17\\hfill \\end{array}[\/latex]<\/p>\r\nProperty 3 states that if two rows or two columns are identical, the determinant equals zero.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}A=\\left[\\left.\\begin{array}{ccc}1&amp; 2&amp; 2\\\\ 2&amp; 2&amp; 2\\\\ -1&amp; 2&amp; 2\\end{array}\\right\\rvert\\begin{array}{c}1\\\\ 2\\\\ -1\\end{array} \\begin{array}{c}2\\\\ 2\\\\ 2\\end{array}\\right]\\hfill \\\\ \\hfill \\\\ \\mathrm{det}\\left(A\\right)=1\\left(2\\right)\\left(2\\right)+2\\left(2\\right)\\left(-1\\right)+2\\left(2\\right)\\left(2\\right)+1\\left(2\\right)\\left(2\\right)-2\\left(2\\right)\\left(1\\right)-2\\left(2\\right)\\left(2\\right)\\hfill \\\\ =4 - 4+8+4 - 4-8=0\\hfill \\end{array}[\/latex]<\/p>\r\nProperty 4 states that if a row or column equals zero, the determinant equals zero. Thus,\r\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{cc}1&amp; 2\\\\ 0&amp; 0\\end{array}\\right],\\mathrm{det}\\left(A\\right)=1\\left(0\\right)-2\\left(0\\right)=0[\/latex]<\/p>\r\nProperty 5 states that the determinant of an inverse matrix [latex]{A}^{-1}[\/latex] is the reciprocal of the determinant [latex]A[\/latex]. Thus,\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}A=\\left[\\begin{array}{cc}1&amp; 2\\\\ 3&amp; 4\\end{array}\\right],\\mathrm{det}\\left(A\\right)=1\\left(4\\right)-3\\left(2\\right)=-2\\hfill \\\\ \\hfill \\\\ {A}^{-1}=\\left[\\begin{array}{cc}-2&amp; 1\\\\ \\frac{3}{2}&amp; -\\frac{1}{2}\\end{array}\\right],\\mathrm{det}\\left({A}^{-1}\\right)=-2\\left(-\\frac{1}{2}\\right)-\\left(\\frac{3}{2}\\right)\\left(1\\right)=-\\frac{1}{2}\\hfill \\end{array}[\/latex]<\/p>\r\nProperty 6 states that if any row or column of a matrix is multiplied by a constant, the determinant is multiplied by the same factor. Thus,\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}A=\\left[\\begin{array}{cc}1&amp; 2\\\\ 3&amp; 4\\end{array}\\right],\\mathrm{det}\\left(A\\right)=1\\left(4\\right)-2\\left(3\\right)=-2\\hfill \\\\ \\hfill \\\\ B=\\left[\\begin{array}{cc}2\\left(1\\right)&amp; 2\\left(2\\right)\\\\ 3&amp; 4\\end{array}\\right],\\mathrm{det}\\left(B\\right)=2\\left(4\\right)-3\\left(4\\right)=-4\\hfill \\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 8: Using Cramer\u2019s Rule and Determinant Properties to Solve a System<\/h3>\r\nFind the solution to the given 3 \u00d7 3 system.\r\n<p style=\"text-align: center;\">[latex]\\begin{gathered}2x+4y+4z=2 \\\\ 3x+7y+7z=-5 \\\\ x+2y+2z=4 \\end{gathered}[\/latex]<\/p>\r\n[reveal-answer q=\"477013\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"477013\"]\r\n\r\nUsing <strong>Cramer\u2019s Rule<\/strong>, we have\r\n<p style=\"text-align: center;\">[latex]D=\\left\\rvert\\begin{array}{ccc}2&amp; 4&amp; 4\\\\ 3&amp; 7&amp; 7\\\\ 1&amp; 2&amp; 2\\end{array}\\right\\rvert[\/latex]<\/p>\r\nNotice that the second and third columns are identical. According to Property 3, the determinant will be zero, so there is either no solution or an infinite number of solutions. We have to perform elimination to find out.\r\n\r\nMultiply equation (3) by \u20132 and add the result to equation (1).\r\n<p style=\"text-align: center;\">[latex]\\begin{align}-2x - 4y - 4x&amp;=-8 \\\\ 2x+4y+4z&amp;=2 \\\\ \\hline 0&amp;=-6\\end{align}[\/latex]<\/p>\r\nObtaining a statement that is a contradiction means that the system has no solution.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Key Concepts<\/h2>\r\n<ul>\r\n \t<li>The determinant for [latex]\\left[\\begin{array}{cc}a&amp; b\\\\ c&amp; d\\end{array}\\right][\/latex] is [latex]ad-bc[\/latex].<\/li>\r\n \t<li>Cramer\u2019s Rule replaces a variable column with the constant column. Solutions are [latex]x=\\frac{{D}_{x}}{D},y=\\frac{{D}_{y}}{D}[\/latex].<\/li>\r\n \t<li>To find the determinant of a 3\u00d73 matrix, augment with the first two columns. Add the three diagonal entries (upper left to lower right) and subtract the three diagonal entries (lower left to upper right).<\/li>\r\n \t<li>To solve a system of three equations in three variables using Cramer\u2019s Rule, replace a variable column with the constant column for each desired solution: [latex]x=\\frac{{D}_{x}}{D},y=\\frac{{D}_{y}}{D},z=\\frac{{D}_{z}}{D}[\/latex].<\/li>\r\n \t<li>Cramer\u2019s Rule is also useful for finding the solution of a system of equations with no solution or infinite solutions.<\/li>\r\n \t<li>Certain properties of determinants are useful for solving problems. For example:\r\n<ul>\r\n \t<li>If the matrix is in upper triangular form, the determinant equals the product of entries down the main diagonal.<\/li>\r\n \t<li>When two rows are interchanged, the determinant changes sign.<\/li>\r\n \t<li>If either two rows or two columns are identical, the determinant equals zero.<\/li>\r\n \t<li>If a matrix contains either a row of zeros or a column of zeros, the determinant equals zero.<\/li>\r\n \t<li>The determinant of an inverse matrix [latex]{A}^{-1}[\/latex] is the reciprocal of the determinant of the matrix [latex]A[\/latex].<\/li>\r\n \t<li>If any row or column is multiplied by a constant, the determinant is multiplied by the same factor.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1674058\" class=\"definition\">\r\n \t<dt>Cramer\u2019s Rule<\/dt>\r\n \t<dd id=\"fs-id1674063\">a method for solving systems of equations that have the same number of equations as variables using determinants<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1674068\" class=\"definition\">\r\n \t<dt>determinant<\/dt>\r\n \t<dd id=\"fs-id1674074\">a number calculated using the entries of a square matrix that determines such information as whether there is a solution to a system of equations<\/dd>\r\n<\/dl>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<p>By the end of this section, you will be able to:<\/p>\n<ul>\n<li>Evaluate 2 \u00d7 2 determinants.<\/li>\n<li>Use Cramer\u2019s Rule to solve a system of equations in two variables.<\/li>\n<li>Evaluate 3 \u00d7 3 determinants.<\/li>\n<li>Use Cramer\u2019s Rule to solve a system of three equations in three variables.<\/li>\n<li>Know the properties of determinants.<\/li>\n<\/ul>\n<\/div>\n<p>We have learned how to solve systems of equations in two variables and three variables, and by multiple methods: substitution, addition, Gaussian elimination, using the inverse of a matrix, and graphing. Some of these methods are easier to apply than others and are more appropriate in certain situations. In this section, we will study two more strategies for solving systems of equations.<\/p>\n<h2>Using Cramer\u2019s Rule to Solve a System of Two Equations in Two Variables<\/h2>\n<h2>Evaluating the Determinant of a 2\u00d72 Matrix<\/h2>\n<p>A determinant is a real number that can be very useful in mathematics because it has multiple applications, such as calculating area, volume, and other quantities. Here, we will use determinants to reveal whether a matrix is invertible by using the entries of a <strong>square matrix<\/strong> to determine whether there is a solution to the system of equations. Perhaps one of the more interesting applications, however, is their use in cryptography. Secure signals or messages are sometimes sent encoded in a matrix. The data can only be decrypted with an <strong>invertible matrix<\/strong> and the determinant. For our purposes, we focus on the determinant as an indication of the invertibility of the matrix. Calculating the determinant of a matrix involves following the specific patterns that are outlined in this section.<\/p>\n<div class=\"textbox shaded\">\n<h3>A General Note: Find the Determinant of a 2 \u00d7 2 Matrix<\/h3>\n<p>The <strong>determinant<\/strong> of a [latex]2\\text{ }\\times \\text{ }2[\/latex] matrix, given<\/p>\n<div style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{cc}a& b\\\\ c& d\\end{array}\\right][\/latex]<\/div>\n<p>is defined as<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181431\/CNX_Precalc_Figure_09_08_0012.jpg\" alt=\"\" width=\"487\" height=\"59\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 1<\/b><\/p>\n<\/div>\n<p>Notice the change in notation. There are several ways to indicate the determinant, including [latex]\\mathrm{det}\\left(A\\right)[\/latex] and replacing the brackets in a matrix with straight lines, [latex]|A|[\/latex].<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 1: Finding the Determinant of a 2 \u00d7 2 Matrix<\/h3>\n<p>Find the determinant of the given matrix.<\/p>\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{cc}5& 2\\\\ -6& 3\\end{array}\\right][\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q149250\">Show Solution<\/span><\/p>\n<div id=\"q149250\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]\\begin{align}\\mathrm{det}\\left(A\\right)&=\\left\\rvert\\begin{array}{cc}5& 2\\\\ -6& 3\\end{array}\\right\\rvert\\hfill \\\\ &=5\\left(3\\right)-\\left(-6\\right)\\left(2\\right)\\hfill \\\\ &=27\\hfill \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm6397\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=6397&theme=oea&iframe_resize_id=ohm6397\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Using Cramer\u2019s Rule to Solve a System of Two Equations in Two Variables<\/h2>\n<p>We will now introduce a final method for solving systems of equations that uses determinants. Known as <strong>Cramer\u2019s Rule<\/strong>, this technique dates back to the middle of the 18th century and is named for its innovator, the Swiss mathematician Gabriel Cramer (1704\u20131752), who introduced it in 1750 in Introduction \u00e0 l&#8217;Analyse des lignes Courbes alg\u00e9briques. Cramer\u2019s Rule is a viable and efficient method for finding solutions to systems with an arbitrary number of unknowns, provided that we have the same number of equations as unknowns.<\/p>\n<p>Cramer\u2019s Rule will give us the unique solution to a system of equations, if it exists. However, if the system has no solution or an infinite number of solutions, this will be indicated by a determinant of zero. To find out if the system is inconsistent or dependent, another method, such as elimination, will have to be used.<\/p>\n<p>To understand Cramer\u2019s Rule, let\u2019s look closely at how we solve systems of linear equations using basic row operations. Consider a system of two equations in two variables.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align}{a}_{1}x+{b}_{1}y&={c}_{1}&&R_{1}\\\\ {a}_{2}x+{b}_{2}y&={c}_{2}&&R_2\\end{align}[\/latex]<\/div>\n<p>We eliminate one variable using row operations and solve for the other. Say that we wish to solve for [latex]x[\/latex]. If equation (2) is multiplied by the opposite of the coefficient of [latex]y[\/latex] in equation (1), equation (1) is multiplied by the coefficient of [latex]y[\/latex] in equation (2), and we add the two equations, the variable [latex]y[\/latex] will be eliminated.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}b_{2}a_{1}x+b_{2}b_{1}y&=b_{2}c_{1} \\\\ \u2212b_{1}a_{2}x\u2212b_{1}b_{2}y&=\u2212b_{1}c_{2} \\\\ \\hline b_{2}a_{1}x\u2212b_{1}a_{2}x&=\u2212b_{2}c_{1}\u2212b_{1}c_{2}\\end{align}[\/latex] [latex]\\begin{align}&\\text{Multiply }R_{1}\\text{ by }b_{2} \\\\ &\\text{Multiply }R_{2}\\text{ by }\u2212b_{2} \\\\ \\text{ } \\end{align}[\/latex]<\/p>\n<p>Now, solve for [latex]x[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{gathered}{b}_{2}{a}_{1}x-{b}_{1}{a}_{2}x={b}_{2}{c}_{1}-{b}_{1}{c}_{2}\\\\ \\hfill \\\\ x\\left({b}_{2}{a}_{1}-{b}_{1}{a}_{2}\\right)={b}_{2}{c}_{1}-{b}_{1}{c}_{2}\\\\ \\hfill \\\\ x=\\frac{{b}_{2}{c}_{1}-{b}_{1}{c}_{2}}{{b}_{2}{a}_{1}-{b}_{1}{a}_{2}}=\\frac{\\left\\rvert\\begin{array}{cc}{c}_{1}& {b}_{1}\\\\ {c}_{2}& {b}_{2}\\end{array}\\right\\rvert}{\\left\\rvert\\begin{array}{cc}{a}_{1}& {b}_{1}\\\\ {a}_{2}& {b}_{2}\\end{array}\\right\\rvert}\\hfill \\end{gathered}[\/latex]<\/div>\n<p>Similarly, to solve for [latex]y[\/latex], we will eliminate [latex]x[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align}a_{2}a_{1}x+a_{2}b_{1}y&=a_{2}c_{1} \\\\\u2212a_{1}a_{2}x\u2212a_{1}b_{2}y&=\u2212a_{1}c_{2} \\\\ \\hline a_{2}b_{1}y\u2212a_{1}b_{2}y&=a_{2}c_{1}\u2212a_{1}c_{2}\\end{align}[\/latex] [latex]\\begin{align}&\\text{Multiply }R_{1}\\text{ by }a_{2} \\\\&\\text{Multiply }R_{2}\\text{ by }\u2212a_{1} \\\\ \\text{ } \\end{align}[\/latex]<\/div>\n<p>Solving for [latex]y[\/latex] gives<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{gathered}{a}_{2}{b}_{1}y-{a}_{1}{b}_{2}y={a}_{2}{c}_{1}-{a}_{1}{c}_{2} \\\\ y\\left({a}_{2}{b}_{1}-{a}_{1}{b}_{2}\\right)={a}_{2}{c}_{1}-{a}_{1}{c}_{2} \\\\ y=\\frac{{a}_{2}{c}_{1}-{a}_{1}{c}_{2}}{{a}_{2}{b}_{1}-{a}_{1}{b}_{2}}=\\frac{{a}_{1}{c}_{2}-{a}_{2}{c}_{1}}{{a}_{1}{b}_{2}-{a}_{2}{b}_{1}}=\\frac{\\left\\rvert\\begin{array}{cc}{a}_{1}& {c}_{1}\\\\ {a}_{2}& {c}_{2}\\end{array}\\right\\rvert}{\\left\\rvert\\begin{array}{cc}{a}_{1}& {b}_{1}\\\\ {a}_{2}& {b}_{2}\\end{array}\\right\\rvert} \\end{gathered}[\/latex]<\/div>\n<p>Notice that the denominator for both [latex]x[\/latex] and [latex]y[\/latex] is the determinant of the coefficient matrix.<\/p>\n<p>We can use these formulas to solve for [latex]x[\/latex] and [latex]y[\/latex], but Cramer\u2019s Rule also introduces new notation:<\/p>\n<ul>\n<li>[latex]D:[\/latex] determinant of the coefficient matrix<\/li>\n<li>[latex]{D}_{x}:[\/latex] determinant of the numerator in the solution of [latex]x[\/latex]\n<div style=\"text-align: center;\">[latex]x=\\frac{{D}_{x}}{D}[\/latex]<\/div>\n<\/li>\n<li>[latex]{D}_{y}:[\/latex] determinant of the numerator in the solution of [latex]y[\/latex]\n<div style=\"text-align: center;\">[latex]y=\\frac{{D}_{y}}{D}[\/latex]<\/div>\n<\/li>\n<\/ul>\n<p>The key to Cramer\u2019s Rule is replacing the variable column of interest with the constant column and calculating the determinants. We can then express [latex]x[\/latex] and [latex]y[\/latex] as a quotient of two determinants.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Cramer\u2019s Rule for 2\u00d72 Systems<\/h3>\n<p><strong>Cramer\u2019s Rule<\/strong> is a method that uses determinants to solve systems of equations that have the same number of equations as variables.<\/p>\n<p>Consider a system of two linear equations in two variables.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}{a}_{1}x+{b}_{1}y={c}_{1}\\\\ {a}_{2}x+{b}_{2}y={c}_{2}\\end{array}[\/latex]<\/p>\n<p>The solution using Cramer\u2019s Rule is given as<\/p>\n<p style=\"text-align: center;\">[latex]x=\\frac{{D}_{x}}{D}=\\frac{\\left\\rvert\\begin{array}{cc}{c}_{1}& {b}_{1}\\\\ {c}_{2}& {b}_{2}\\end{array}\\right\\rvert}{\\left\\rvert\\begin{array}{cc}{a}_{1}& {b}_{1}\\\\ {a}_{2}& {b}_{2}\\end{array}\\right\\rvert},D\\ne 0;\\text{ }\\text{ }y=\\frac{{D}_{y}}{D}=\\frac{\\left\\rvert\\begin{array}{cc}{a}_{1}& {c}_{1}\\\\ {a}_{2}& {c}_{2}\\end{array}\\right\\rvert}{\\left\\rvert\\begin{array}{cc}{a}_{1}& {b}_{1}\\\\ {a}_{2}& {b}_{2}\\end{array}\\right\\rvert},D\\ne 0[\/latex].<\/p>\n<p>If we are solving for [latex]x[\/latex], the [latex]x[\/latex] column is replaced with the constant column. If we are solving for [latex]y[\/latex], the [latex]y[\/latex] column is replaced with the constant column.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 2: Using Cramer\u2019s Rule to Solve a 2 \u00d7 2 System<\/h3>\n<p>Solve the following [latex]2\\text{ }\\times \\text{ }2[\/latex] system using Cramer\u2019s Rule.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}12x+3y&=15\\\\ 2x - 3y&=13\\end{align}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q472105\">Show Solution<\/span><\/p>\n<div id=\"q472105\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solve for [latex]x[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]x=\\frac{{D}_{x}}{D}=\\frac{\\left\\rvert\\begin{array}{rr}\\hfill 15& \\hfill 3\\\\ \\hfill 13& \\hfill -3\\end{array}\\right\\rvert}{\\left\\rvert\\begin{array}{rr}\\hfill 12& \\hfill 3\\\\ \\hfill 2& \\hfill -3\\end{array}\\right\\rvert}=\\frac{-45 - 39}{-36 - 6}=\\frac{-84}{-42}=2[\/latex]<\/p>\n<p>Solve for [latex]y[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]y=\\frac{{D}_{y}}{D}=\\frac{\\left\\rvert\\begin{array}{rr}\\hfill 12& \\hfill 15\\\\ \\hfill 2& \\hfill 13\\end{array}\\right\\rvert}{\\left\\rvert\\begin{array}{rr}\\hfill 12& \\hfill 3\\\\ \\hfill 2& \\hfill -3\\end{array}\\right\\rvert}=\\frac{156 - 30}{-36 - 6}=-\\frac{126}{42}=-3[\/latex]<\/p>\n<p>The solution is [latex]\\left(2,-3\\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Use Cramer\u2019s Rule to solve the 2 \u00d7 2 system of equations.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{gathered}x+2y=-11 \\\\ -2x+y=-13 \\end{gathered}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q498348\">Show Solution<\/span><\/p>\n<div id=\"q498348\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left(3,-7\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm174705\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=174705&theme=oea&iframe_resize_id=ohm174705\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Using Cramer\u2019s Rule to Solve a System of Three Equations in Three Variables<\/h2>\n<h2>Evaluating the Determinant of a 3 \u00d7 3 Matrix<\/h2>\n<p>Finding the determinant of a 2\u00d72 matrix is straightforward, but finding the determinant of a 3\u00d73 matrix is more complicated. One method is to augment the 3\u00d73 matrix with a repetition of the first two columns, giving a 3\u00d75 matrix. Then we calculate the sum of the products of entries <em>down<\/em> each of the three diagonals (upper left to lower right), and subtract the products of entries <em>up<\/em> each of the three diagonals (lower left to upper right). This is more easily understood with a visual and an example.<\/p>\n<p>Find the <strong>determinant<\/strong> of the 3\u00d73 matrix.<\/p>\n<div style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{ccc}{a}_{1}& {b}_{1}& {c}_{1}\\\\ {a}_{2}& {b}_{2}& {c}_{2}\\\\ {a}_{3}& {b}_{3}& {c}_{3}\\end{array}\\right][\/latex]<\/div>\n<ol>\n<li>Augment [latex]A[\/latex] with the first two columns.\n<div style=\"text-align: center;\">[latex]\\mathrm{det}\\left(A\\right)=\\left\\rvert\\begin{array}{ccc}{a}_{1}& {b}_{1}& {c}_{1}\\\\ {a}_{2}& {b}_{2}& {c}_{2}\\\\ {a}_{3}& {b}_{3}& {c}_{3}\\end{array}\\right\\rvert \\left.\\begin{array}{c}{a}_{1}\\\\ {a}_{2}\\\\ {a}_{3}\\end{array}\\begin{array}{c}{b}_{1}\\\\ {b}_{2}\\\\ {b}_{3}\\end{array}\\right\\rvert[\/latex]<\/div>\n<\/li>\n<li>From upper left to lower right: Multiply the entries down the first diagonal. Add the result to the product of entries down the second diagonal. Add this result to the product of the entries down the third diagonal.<\/li>\n<li>From lower left to upper right: Subtract the product of entries up the first diagonal. From this result subtract the product of entries up the second diagonal. From this result, subtract the product of entries up the third diagonal.<\/li>\n<\/ol>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181433\/CNX_Precalc_Figure_09_08_0022.jpg\" alt=\"\" width=\"487\" height=\"89\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 2<\/b><\/p>\n<\/div>\n<p>The algebra is as follows:<\/p>\n<div style=\"text-align: center;\">[latex]|A|={a}_{1}{b}_{2}{c}_{3}+{b}_{1}{c}_{2}{a}_{3}+{c}_{1}{a}_{2}{b}_{3}-{a}_{3}{b}_{2}{c}_{1}-{b}_{3}{c}_{2}{a}_{1}-{c}_{3}{a}_{2}{b}_{1}[\/latex]<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 3: Finding the Determinant of a 3 \u00d7 3 Matrix<\/h3>\n<p>Find the determinant of the 3 \u00d7 3 matrix given<\/p>\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{ccc}0& 2& 1\\\\ 3& -1& 1\\\\ 4& 0& 1\\end{array}\\right][\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q915069\">Show Solution<\/span><\/p>\n<div id=\"q915069\" class=\"hidden-answer\" style=\"display: none\">\n<p>Augment the matrix with the first two columns and then follow the formula. Thus,<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}|A|&=\\left\\rvert\\begin{array}{ccc}0& 2& 1\\\\ 3& -1& 1\\\\ 4& 0& 1\\end{array}\\right\\rvert\\left.\\begin{array}{c}0 & 2\\\\ 3 & -1\\\\ 4 & 0\\end{array}\\right\\rvert\\hfill \\\\ &=0\\left(-1\\right)\\left(1\\right)+2\\left(1\\right)\\left(4\\right)+1\\left(3\\right)\\left(0\\right)-4\\left(-1\\right)\\left(1\\right)-0\\left(1\\right)\\left(0\\right)-1\\left(3\\right)\\left(2\\right)\\hfill \\\\ &=0+8+0+4 - 0-6\\hfill \\\\ &=6\\hfill \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Find the determinant of the 3 \u00d7 3 matrix.<\/p>\n<p style=\"text-align: center;\">[latex]\\mathrm{det}\\left(A\\right)=\\left\\rvert\\begin{array}{ccc}1& -3& 7\\\\ 1& 1& 1\\\\ 1& -2& 3\\end{array}\\right\\rvert[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q612489\">Show Solution<\/span><\/p>\n<div id=\"q612489\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]-10[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm19398\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=19398&theme=oea&iframe_resize_id=ohm19398\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<h3>Can we use the same method to find the determinant of a larger matrix?<\/h3>\n<p><em>Yes, but for larger matrices it is best to use a graphing utility or computer software.<\/em><\/p>\n<\/div>\n<h2>Using Cramer\u2019s Rule to Solve a System of Three Equations in Three Variables<\/h2>\n<p>Now that we can find the <strong>determinant<\/strong> of a 3 \u00d7 3 matrix, we can apply <strong>Cramer\u2019s Rule<\/strong> to solve a <strong>system of three equations in three variables<\/strong>. Cramer\u2019s Rule is straightforward, following a pattern consistent with Cramer\u2019s Rule for 2 \u00d7 2 matrices. As the order of the matrix increases to 3 \u00d7 3, however, there are many more calculations required.<\/p>\n<p>When we calculate the determinant to be zero, Cramer\u2019s Rule gives no indication as to whether the system has no solution or an infinite number of solutions. To find out, we have to perform elimination on the system.<\/p>\n<p>Consider a 3 \u00d7 3 system of equations.<\/p>\n<div id=\"attachment_12665\" style=\"width: 253px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/145\/2017\/01\/06215037\/Screen-Shot-2017-01-06-at-1.50.12-PM.png\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-12665\" class=\"size-full wp-image-12665\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181435\/Screen-Shot-2017-01-06-at-1.50.12-PM.png\" alt=\"\" width=\"243\" height=\"136\" \/><\/a><\/p>\n<p id=\"caption-attachment-12665\" class=\"wp-caption-text\"><strong>Figure 3<\/strong><\/p>\n<\/div>\n<div style=\"text-align: center;\">[latex]x=\\frac{{D}_{x}}{D},y=\\frac{{D}_{y}}{D},z=\\frac{{D}_{z}}{D},D\\ne 0[\/latex]<\/div>\n<p>where<\/p>\n<div id=\"attachment_12667\" style=\"width: 866px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/145\/2017\/01\/06215135\/Screen-Shot-2017-01-06-at-1.51.06-PM.png\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-12667\" class=\"size-full wp-image-12667\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181437\/Screen-Shot-2017-01-06-at-1.51.06-PM.png\" alt=\"\" width=\"856\" height=\"128\" \/><\/a><\/p>\n<p id=\"caption-attachment-12667\" class=\"wp-caption-text\"><strong>Figure 4<\/strong><\/p>\n<\/div>\n<p>If we are writing the determinant [latex]{D}_{x}[\/latex], we replace the [latex]x[\/latex] column with the constant column. If we are writing the determinant [latex]{D}_{y}[\/latex], we replace the [latex]y[\/latex] column with the constant column. If we are writing the determinant [latex]{D}_{z}[\/latex], we replace the [latex]z[\/latex] column with the constant column. Always check the answer.<\/p>\n<div class=\"textbox shaded\">\n<h3>Example 4: Solving a 3 \u00d7 3 System Using Cramer\u2019s Rule<\/h3>\n<p>Find the solution to the given 3 \u00d7 3 system using Cramer\u2019s Rule.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{gathered}x+y-z=6\\\\ 3x - 2y+z=-5\\\\ x+3y - 2z=14\\end{gathered}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q180631\">Show Solution<\/span><\/p>\n<div id=\"q180631\" class=\"hidden-answer\" style=\"display: none\">\n<p>Use Cramer\u2019s Rule.<\/p>\n<p style=\"text-align: center;\">[latex]D=\\left\\rvert\\begin{array}{ccc}1& 1& -1\\\\ 3& -2& 1\\\\ 1& 3& -2\\end{array}\\right\\rvert\\text{, }{D}_{x}=\\left\\rvert\\begin{array}{ccc}6& 1& -1\\\\ -5& -2& 1\\\\ 14& 3& -2\\end{array}\\right\\rvert\\text{, }{D}_{y}=\\left\\rvert\\begin{array}{ccc}1& 6& -1\\\\ 3& -5& 1\\\\ 1& 14& -2\\end{array}\\right\\rvert\\text{, }{D}_{z}=\\left\\rvert\\begin{array}{ccc}1& 1& 6\\\\ 3& -2& -5\\\\ 1& 3& 14\\end{array}\\right\\rvert[\/latex]<\/p>\n<p>Then,<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}x&=\\frac{{D}_{x}}{D}=\\frac{-3}{-3}=1\\hfill \\\\ y&=\\frac{{D}_{y}}{D}=\\frac{-9}{-3}=3\\hfill \\\\ z&=\\frac{{D}_{z}}{D}=\\frac{6}{-3}=-2\\hfill \\end{align}[\/latex]<\/p>\n<p>The solution is [latex]\\left(1,3,-2\\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Use Cramer\u2019s Rule to solve the 3 \u00d7 3 matrix.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{gathered} x - 3y+7z=13\\\\ x+y+z=1\\\\ x - 2y+3z=4\\end{gathered}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q700599\">Show Solution<\/span><\/p>\n<div id=\"q700599\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left(-2,\\frac{3}{5},\\frac{12}{5}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 5: Using Cramer\u2019s Rule to Solve an Inconsistent System<\/h3>\n<p>Solve the system of equations using Cramer\u2019s Rule.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{gathered}3x - 2y=4 \\\\ 6x - 4y=0\\end{gathered}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q966288\">Show Solution<\/span><\/p>\n<div id=\"q966288\" class=\"hidden-answer\" style=\"display: none\">\n<p>We begin by finding the determinants [latex]D,{D}_{x},\\text{and }{D}_{y}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]D=\\left\\rvert\\begin{array}{cc}3& -2\\\\ 6& -4\\end{array}\\right\\rvert=3\\left(-4\\right)-6\\left(-2\\right)=0[\/latex]<\/p>\n<p>We know that a determinant of zero means that either the system has no solution or it has an infinite number of solutions. To see which one, we use the process of elimination. Our goal is to eliminate one of the variables.<\/p>\n<ol>\n<li>Multiply equation (1) by [latex]-2[\/latex].<\/li>\n<li>Add the result to equation [latex]\\left(2\\right)[\/latex].<\/li>\n<\/ol>\n<p style=\"text-align: center;\">[latex]\\begin{align}\u22126x+4y&=\u22128 \\\\ 6x\u22124y&=0 \\\\ \\hline0&=-8\\end{align}[\/latex]<\/p>\n<p>We obtain the equation [latex]0=-8[\/latex], which is false. Therefore, the system has no solution. Graphing the system reveals two parallel lines.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181440\/CNX_Precalc_Figure_09_08_0032.jpg\" alt=\"Graph of two parallel lines with the equations y=three-halves x and y=three-halves x minus 2.\" width=\"487\" height=\"441\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 5<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 6: Use Cramer\u2019s Rule to Solve a Dependent System<\/h3>\n<p>Solve the system with an infinite number of solutions.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{gathered} x - 2y+3z=0\\\\ 3x+y - 2z=0 \\\\ 2x - 4y+6z=0 \\end{gathered}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q282282\">Show Solution<\/span><\/p>\n<div id=\"q282282\" class=\"hidden-answer\" style=\"display: none\">\n<p>Let\u2019s find the determinant first. Set up a matrix augmented by the first two columns.<\/p>\n<p style=\"text-align: center;\">[latex]\\left\\rvert\\begin{array}{rrr}\\hfill 1& \\hfill -2& \\hfill 3\\\\ \\hfill 3& \\hfill 1& \\hfill -2\\\\ \\hfill 2& \\hfill -4& \\hfill 6\\end{array}\\right\\rvert\\left.\\begin{array}{rr}\\hfill 1& \\hfill -2\\\\ \\hfill 3& \\hfill 1\\\\ \\hfill 2& \\hfill -4\\end{array}\\right\\rvert[\/latex]<\/p>\n<p>Then,<\/p>\n<p style=\"text-align: center;\">[latex]1\\left(1\\right)\\left(6\\right)+\\left(-2\\right)\\left(-2\\right)\\left(2\\right)+3\\left(3\\right)\\left(-4\\right)-2\\left(1\\right)\\left(3\\right)-\\left(-4\\right)\\left(-2\\right)\\left(1\\right)-6\\left(3\\right)\\left(-2\\right)=0[\/latex]<\/p>\n<p>As the determinant equals zero, there is either no solution or an infinite number of solutions. We have to perform elimination to find out.<\/p>\n<ol>\n<li>Multiply equation (1) by [latex]-2[\/latex] and add the result to equation (3):\n<div style=\"text-align: center;\">[latex]\\begin{align} -2x+4y - 6z&=0\\\\ 2x - 4y+6z&=0\\\\ \\hline 0&=0 \\end{align}[\/latex]<\/div>\n<\/li>\n<li>Obtaining an answer of [latex]0=0[\/latex], a statement that is always true, means that the system has an infinite number of solutions. Graphing the system, we can see that two of the planes are the same and they both intersect the third plane on a line.<\/li>\n<\/ol>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181442\/CNX_Precalc_Figure_09_08_0052.jpg\" alt=\"Two planes intersecting a third plane. One plane's equation is x minus 2y plus 3z equals zero. The second plane's equation is 2x minus 4y plus 6z equals zero. The third plane's equation is 3x plus y plus 2z equals zero.\" width=\"487\" height=\"214\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 6<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm174706\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=174706&theme=oea&iframe_resize_id=ohm174706\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Understanding Properties of Determinants<\/h2>\n<p>There are many <strong>properties of determinants<\/strong>. Listed here are some properties that may be helpful in calculating the determinant of a matrix.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Properties of Determinants<\/h3>\n<ol>\n<li>If the matrix is in upper triangular form, the determinant equals the product of entries down the main diagonal.<\/li>\n<li>When two rows are interchanged, the determinant changes sign.<\/li>\n<li>If either two rows or two columns are identical, the determinant equals zero.<\/li>\n<li>If a matrix contains either a row of zeros or a column of zeros, the determinant equals zero.<\/li>\n<li>The determinant of an inverse matrix [latex]{A}^{-1}[\/latex] is the reciprocal of the determinant of the matrix [latex]A[\/latex].<\/li>\n<li>If any row or column is multiplied by a constant, the determinant is multiplied by the same factor.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 7: Illustrating Properties of Determinants<\/h3>\n<p>Illustrate each of the properties of determinants.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q123702\">Show Solution<\/span><\/p>\n<div id=\"q123702\" class=\"hidden-answer\" style=\"display: none\">\n<p>Property 1 states that if the matrix is in upper triangular form, the determinant is the product of the entries down the main diagonal.<\/p>\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{rrr}\\hfill 1& \\hfill 2& \\hfill 3\\\\ \\hfill 0& \\hfill 2& \\hfill 1\\\\ \\hfill 0& \\hfill 0& \\hfill -1\\end{array}\\right][\/latex]<\/p>\n<p>Augment [latex]A[\/latex] with the first two columns.<\/p>\n<p style=\"text-align: center;\">[latex]A=\\left[\\left.\\begin{array}{ccc}1& 2& 3\\\\ 0& 2& 1\\\\ 0& 0& -1\\end{array}\\right\\rvert\\begin{array}{c}1\\\\ 0\\\\ 0\\end{array}\\begin{array}{c}2\\\\ 2\\\\ 0\\end{array}\\right][\/latex]<\/p>\n<p>Then<\/p>\n<p style=\"text-align: center;\">[latex]\\mathrm{det}\\left(A\\right)=1\\left(2\\right)\\left(-1\\right)+2\\left(1\\right)\\left(0\\right)+3\\left(0\\right)\\left(0\\right)-0\\left(2\\right)\\left(3\\right)-0\\left(1\\right)\\left(1\\right)+1\\left(0\\right)\\left(2\\right)=-2[\/latex]<\/p>\n<p>Property 2 states that interchanging rows changes the sign. Given<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\begin{array}{l}\\\\ A=\\left[\\begin{array}{cc}-1& 5\\\\ 4& -3\\end{array}\\right],\\mathrm{det}\\left(A\\right)=\\left(-1\\right)\\left(-3\\right)-\\left(4\\right)\\left(5\\right)=3 - 20=-17\\end{array}\\hfill \\\\ \\hfill \\\\ B=\\left[\\begin{array}{cc}4& -3\\\\ -1& 5\\end{array}\\right],\\mathrm{det}\\left(B\\right)=\\left(4\\right)\\left(5\\right)-\\left(-1\\right)\\left(-3\\right)=20 - 3=17\\hfill \\end{array}[\/latex]<\/p>\n<p>Property 3 states that if two rows or two columns are identical, the determinant equals zero.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}A=\\left[\\left.\\begin{array}{ccc}1& 2& 2\\\\ 2& 2& 2\\\\ -1& 2& 2\\end{array}\\right\\rvert\\begin{array}{c}1\\\\ 2\\\\ -1\\end{array} \\begin{array}{c}2\\\\ 2\\\\ 2\\end{array}\\right]\\hfill \\\\ \\hfill \\\\ \\mathrm{det}\\left(A\\right)=1\\left(2\\right)\\left(2\\right)+2\\left(2\\right)\\left(-1\\right)+2\\left(2\\right)\\left(2\\right)+1\\left(2\\right)\\left(2\\right)-2\\left(2\\right)\\left(1\\right)-2\\left(2\\right)\\left(2\\right)\\hfill \\\\ =4 - 4+8+4 - 4-8=0\\hfill \\end{array}[\/latex]<\/p>\n<p>Property 4 states that if a row or column equals zero, the determinant equals zero. Thus,<\/p>\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{cc}1& 2\\\\ 0& 0\\end{array}\\right],\\mathrm{det}\\left(A\\right)=1\\left(0\\right)-2\\left(0\\right)=0[\/latex]<\/p>\n<p>Property 5 states that the determinant of an inverse matrix [latex]{A}^{-1}[\/latex] is the reciprocal of the determinant [latex]A[\/latex]. Thus,<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}A=\\left[\\begin{array}{cc}1& 2\\\\ 3& 4\\end{array}\\right],\\mathrm{det}\\left(A\\right)=1\\left(4\\right)-3\\left(2\\right)=-2\\hfill \\\\ \\hfill \\\\ {A}^{-1}=\\left[\\begin{array}{cc}-2& 1\\\\ \\frac{3}{2}& -\\frac{1}{2}\\end{array}\\right],\\mathrm{det}\\left({A}^{-1}\\right)=-2\\left(-\\frac{1}{2}\\right)-\\left(\\frac{3}{2}\\right)\\left(1\\right)=-\\frac{1}{2}\\hfill \\end{array}[\/latex]<\/p>\n<p>Property 6 states that if any row or column of a matrix is multiplied by a constant, the determinant is multiplied by the same factor. Thus,<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}A=\\left[\\begin{array}{cc}1& 2\\\\ 3& 4\\end{array}\\right],\\mathrm{det}\\left(A\\right)=1\\left(4\\right)-2\\left(3\\right)=-2\\hfill \\\\ \\hfill \\\\ B=\\left[\\begin{array}{cc}2\\left(1\\right)& 2\\left(2\\right)\\\\ 3& 4\\end{array}\\right],\\mathrm{det}\\left(B\\right)=2\\left(4\\right)-3\\left(4\\right)=-4\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 8: Using Cramer\u2019s Rule and Determinant Properties to Solve a System<\/h3>\n<p>Find the solution to the given 3 \u00d7 3 system.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{gathered}2x+4y+4z=2 \\\\ 3x+7y+7z=-5 \\\\ x+2y+2z=4 \\end{gathered}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q477013\">Show Solution<\/span><\/p>\n<div id=\"q477013\" class=\"hidden-answer\" style=\"display: none\">\n<p>Using <strong>Cramer\u2019s Rule<\/strong>, we have<\/p>\n<p style=\"text-align: center;\">[latex]D=\\left\\rvert\\begin{array}{ccc}2& 4& 4\\\\ 3& 7& 7\\\\ 1& 2& 2\\end{array}\\right\\rvert[\/latex]<\/p>\n<p>Notice that the second and third columns are identical. According to Property 3, the determinant will be zero, so there is either no solution or an infinite number of solutions. We have to perform elimination to find out.<\/p>\n<p>Multiply equation (3) by \u20132 and add the result to equation (1).<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}-2x - 4y - 4x&=-8 \\\\ 2x+4y+4z&=2 \\\\ \\hline 0&=-6\\end{align}[\/latex]<\/p>\n<p>Obtaining a statement that is a contradiction means that the system has no solution.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Key Concepts<\/h2>\n<ul>\n<li>The determinant for [latex]\\left[\\begin{array}{cc}a& b\\\\ c& d\\end{array}\\right][\/latex] is [latex]ad-bc[\/latex].<\/li>\n<li>Cramer\u2019s Rule replaces a variable column with the constant column. Solutions are [latex]x=\\frac{{D}_{x}}{D},y=\\frac{{D}_{y}}{D}[\/latex].<\/li>\n<li>To find the determinant of a 3\u00d73 matrix, augment with the first two columns. Add the three diagonal entries (upper left to lower right) and subtract the three diagonal entries (lower left to upper right).<\/li>\n<li>To solve a system of three equations in three variables using Cramer\u2019s Rule, replace a variable column with the constant column for each desired solution: [latex]x=\\frac{{D}_{x}}{D},y=\\frac{{D}_{y}}{D},z=\\frac{{D}_{z}}{D}[\/latex].<\/li>\n<li>Cramer\u2019s Rule is also useful for finding the solution of a system of equations with no solution or infinite solutions.<\/li>\n<li>Certain properties of determinants are useful for solving problems. For example:\n<ul>\n<li>If the matrix is in upper triangular form, the determinant equals the product of entries down the main diagonal.<\/li>\n<li>When two rows are interchanged, the determinant changes sign.<\/li>\n<li>If either two rows or two columns are identical, the determinant equals zero.<\/li>\n<li>If a matrix contains either a row of zeros or a column of zeros, the determinant equals zero.<\/li>\n<li>The determinant of an inverse matrix [latex]{A}^{-1}[\/latex] is the reciprocal of the determinant of the matrix [latex]A[\/latex].<\/li>\n<li>If any row or column is multiplied by a constant, the determinant is multiplied by the same factor.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1674058\" class=\"definition\">\n<dt>Cramer\u2019s Rule<\/dt>\n<dd id=\"fs-id1674063\">a method for solving systems of equations that have the same number of equations as variables using determinants<\/dd>\n<\/dl>\n<dl id=\"fs-id1674068\" class=\"definition\">\n<dt>determinant<\/dt>\n<dd id=\"fs-id1674074\">a number calculated using the entries of a square matrix that determines such information as whether there is a solution to a system of equations<\/dd>\n<\/dl>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-14608\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: OpenStax College. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":8,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-14608","chapter","type-chapter","status-publish","hentry"],"part":14549,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/14608","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":6,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/14608\/revisions"}],"predecessor-version":[{"id":15854,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/14608\/revisions\/15854"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/14549"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/14608\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=14608"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=14608"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=14608"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=14608"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}