{"id":401,"date":"2019-07-15T22:44:45","date_gmt":"2019-07-15T22:44:45","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/chapter\/add-and-subtract-matrices\/"},"modified":"2019-07-15T22:44:45","modified_gmt":"2019-07-15T22:44:45","slug":"add-and-subtract-matrices","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/chapter\/add-and-subtract-matrices\/","title":{"raw":"Adding and Subtracting Matrices","rendered":"Adding and Subtracting Matrices"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n \t<li>Determine the dimensions of a matrix.<\/li>\n \t<li>Add and subtract two matrices.<\/li>\n<\/ul>\n<\/div>\nTo solve a problem like the one described for the soccer teams, we can use a <strong>matrix<\/strong>, which is a rectangular array of numbers. A <strong>row<\/strong> in a matrix is a set of numbers that are aligned horizontally. A <strong>column<\/strong> in a matrix is a set of numbers that are aligned vertically. Each number is an <strong>entry<\/strong>, sometimes called an element, of the matrix. Matrices (plural) are enclosed in [ ] or ( ) and are usually named with capital letters. For example, three matrices named [latex]A,B,\\text{}[\/latex] and [latex]C[\/latex] are shown below.\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{cc}1&amp; 2\\\\ 3&amp; 4\\end{array}\\right],B=\\left[\\begin{array}{ccc}\\hfill 1&amp; \\hfill 2&amp; \\hfill 7\\\\ \\hfill 0&amp; \\hfill -5&amp; \\hfill 6\\\\ \\hfill 7&amp; \\hfill 8&amp; \\hfill 2\\end{array}\\right],C=\\left[\\begin{array}{c}\\hfill -1\\\\ \\hfill 0\\\\ \\hfill 3\\end{array}\\begin{array}{c}3\\\\ 2\\\\ 1\\end{array}\\right][\/latex]<\/p>\n\n<h2>Describing Matrices<\/h2>\nA matrix is often referred to by its size or dimensions: [latex]\\text{ }m\\text{ }\\times \\text{ }n\\text{ }[\/latex] indicating [latex]m[\/latex] rows and [latex]n[\/latex] columns. Matrix entries are defined first by row and then by column. For example, to locate the entry in matrix [latex]A[\/latex] identified as [latex]{a}_{ij},\\text{}[\/latex] we look for the entry in row [latex]i,\\text{}[\/latex] column [latex]j[\/latex]. In matrix [latex]A[\/latex] shown below, the entry in row 2, column 3 is [latex]{a}_{23}[\/latex].\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{ccc}{a}_{11}&amp; {a}_{12}&amp; {a}_{13}\\\\ {a}_{21}&amp; {a}_{22}&amp; {a}_{23}\\\\ {a}_{31}&amp; {a}_{32}&amp; {a}_{33}\\end{array}\\right][\/latex]<\/p>\n\n<div class=\"textbox examples\">\n<h3>how are those subscripts pronounced?<\/h3>\nSince the subscripts in a matrix refer to row and column entry locations, the numbers are pronounced distinctly from one another. To refer to the 1st row, third column entry in the matrix [latex]A[\/latex] above, we say \"[latex]a \\text{ one }\\text{three}[\/latex].\"\n\n<\/div>\nA <strong>square matrix<\/strong> is a matrix with dimensions [latex]\\text{ }n\\text{ }\\times \\text{ }n,\\text{}[\/latex] meaning that it has the same number of rows as columns. The [latex]3\\times 3[\/latex] matrix above is an example of a square matrix.\n\nA <strong>row matrix<\/strong> is a matrix consisting of one row with dimensions [latex]1\\text{ }\\times \\text{ }n[\/latex].\n<p style=\"text-align: center;\">[latex]\\left[\\begin{array}{ccc}{a}_{11}&amp; {a}_{12}&amp; {a}_{13}\\end{array}\\right][\/latex]<\/p>\nA <strong>column matrix<\/strong> is a matrix consisting of one column with dimensions [latex]m\\text{ }\\times \\text{ }1[\/latex].\n<p style=\"text-align: center;\">[latex]\\left[\\begin{array}{c}{a}_{11}\\\\ {a}_{21}\\\\ {a}_{31}\\end{array}\\right][\/latex]<\/p>\nA matrix may be used to represent a system of equations. In these cases, the numbers represent the coefficients of the variables in the system. Matrices often make solving systems of equations easier because they are not encumbered with variables. We will investigate this idea further in the next section, but first we will look at basic <strong>matrix operations<\/strong>.\n<div class=\"textbox\">\n<h3>A General Note: Matrices<\/h3>\nA <strong>matrix<\/strong> is a rectangular array of numbers that is usually named by a capital letter: [latex]A,B,C,\\text{}[\/latex] and so on. Each entry in a matrix is referred to as [latex]{a}_{ij}[\/latex], such that [latex]i[\/latex] represents the row and [latex]j[\/latex] represents the column. Matrices are often referred to by their dimensions: [latex]m\\times n[\/latex] indicating [latex]m[\/latex] rows and [latex]n[\/latex] columns.\n\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Dimensions of the Given Matrix and Locating Entries<\/h3>\nGiven matrix [latex]A:[\/latex]\n<ol>\n \t<li>What are the dimensions of matrix [latex]A?[\/latex]<\/li>\n \t<li>What are the entries at [latex]{a}_{31}[\/latex] and [latex]{a}_{22}?[\/latex]<\/li>\n<\/ol>\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{rrrr}\\hfill 2&amp; \\hfill &amp; \\hfill 1&amp; \\hfill 0\\\\ \\hfill 2&amp; \\hfill &amp; \\hfill 4&amp; \\hfill 7\\\\ \\hfill 3&amp; \\hfill &amp; \\hfill 1&amp; \\hfill -2\\end{array}\\right][\/latex]<\/p>\n[reveal-answer q=\"832047\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"832047\"]\n<ol>\n \t<li>The dimensions are [latex]\\text{ }3\\text{ }\\times \\text{ }3\\text{ }[\/latex] because there are three rows and three columns.<\/li>\n \t<li>Entry [latex]{a}_{31}[\/latex] is the number at row 3, column 1 which is 3. The entry [latex]{a}_{22}[\/latex] is the number at row 2, column 2 which is 4. Remember, the row comes first, then the column.<\/li>\n<\/ol>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\n[ohm_question]6388[\/ohm_question]\n\n<\/div>\n<h2>Adding and Subtracting Matrices<\/h2>\nWe use matrices to list data or to represent systems. Because the entries are numbers, we can perform operations on matrices. We add or subtract matrices by adding or subtracting corresponding entries.\n\nIn order to do this, the entries must correspond. Therefore, <em>addition and subtraction of matrices is only possible when the matrices have the same dimensions<\/em>. We can add or subtract a [latex]\\text{ }3\\text{ }\\times \\text{ }3\\text{ }[\/latex] matrix and another [latex]\\text{ }3\\text{ }\\times \\text{ }3\\text{ }[\/latex] matrix, but we cannot add or subtract a [latex]\\text{ }2\\text{ }\\times \\text{ }3\\text{ }[\/latex] matrix and a [latex]\\text{ }3\\text{ }\\times \\text{ }3\\text{ }[\/latex] matrix because some entries in one matrix will not have a corresponding entry in the other matrix.\n<div class=\"textbox\">\n<h3>A General Note: Adding and Subtracting Matrices<\/h3>\nGiven matrices [latex]A[\/latex] and [latex]B[\/latex] of like dimensions, addition and subtraction of [latex]A[\/latex] and [latex]B[\/latex] will produce matrix [latex]C[\/latex] or&nbsp;matrix [latex]D[\/latex] of the same dimension.\n<p style=\"text-align: center;\">[latex]A+B=C\\text{ such that }{a}_{ij}+{b}_{ij}={c}_{ij}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]A-B=D\\text{ such that }{a}_{ij}-{b}_{ij}={d}_{ij}[\/latex]<\/p>\nMatrix addition is commutative.\n<p style=\"text-align: center;\">[latex]A+B=B+A[\/latex]<\/p>\nIt is also associative.\n<p style=\"text-align: center;\">[latex]\\left(A+B\\right)+C=A+\\left(B+C\\right)[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Example: Finding the Sum of Matrices<\/h3>\nFind the sum of [latex]A[\/latex] and [latex]B \\text{}[\/latex] given\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{cc}a&amp; b\\\\ c&amp; d\\end{array}\\right]\\text{ and }B=\\left[\\begin{array}{cc}e&amp; f\\\\ g&amp; h\\end{array}\\right][\/latex]<\/p>\n[reveal-answer q=\"3634\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"3634\"]\n\nAdd corresponding entries.\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}A+B &amp; =\\left[\\begin{array}{cc}a&amp; b\\\\ c&amp; d\\end{array}\\right]+\\left[\\begin{array}{cc}e&amp; f\\\\ g&amp; h\\end{array}\\right]\\hfill \\\\ &amp; =\\left[\\begin{array}{ccc}a+e&amp; &amp; b+f\\\\ c+g&amp; &amp; d+h\\end{array}\\right]\\hfill \\end{array}[\/latex]<\/p>\n[\/hidden-answer]\n\n[ohm_question]1079[\/ohm_question]\n\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Adding Matrix <em>A <\/em>and Matrix B<\/h3>\nFind the sum of [latex]A[\/latex] and [latex]B[\/latex].\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{cc}4&amp; 1\\\\ 3&amp; 2\\end{array}\\right]\\text{ and }B=\\left[\\begin{array}{cc}5&amp; 9\\\\ 0&amp; 7\\end{array}\\right][\/latex]<\/p>\n[reveal-answer q=\"512431\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"512431\"]\n\nAdd corresponding entries. Add the entry in row 1, column 1, [latex]{a}_{11},\\text{}[\/latex] of matrix [latex]A[\/latex] to the entry in row 1, column 1, [latex]{b}_{11}[\/latex], of [latex]B[\/latex]. Continue the pattern until all entries have been added.\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll}A+B &amp; =\\left[\\begin{array}{cc}4&amp; 1\\\\ 3&amp; 2\\end{array}\\right]+\\left[\\begin{array}{cc}5&amp; 9\\\\ 0&amp; 7\\end{array}\\right]\\hfill \\\\ &amp; =\\left[\\begin{array}{ccc}4+5&amp; &amp; 1+9\\\\ 3+0&amp; &amp; 2+7\\end{array}\\right]\\hfill \\\\ &amp; =\\left[\\begin{array}{cc}9&amp; 10\\\\ 3&amp; 9\\end{array}\\right]\\hfill \\end{array}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Difference of Two Matrices<\/h3>\nFind the difference of [latex]A[\/latex] and [latex]B[\/latex].\n[latex]A=\\left[\\begin{array}{cc}\\hfill -2&amp; \\hfill 3\\\\ \\hfill 0&amp; \\hfill 1\\end{array}\\right]\\text{ and }B=\\left[\\begin{array}{cc}8&amp; 1\\\\ 5&amp; 4\\end{array}\\right][\/latex]\n\n[reveal-answer q=\"83802\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"83802\"]\n\nWe subtract the corresponding entries of each matrix.\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll}A-B &amp; =\\left[\\begin{array}{rr}\\hfill -2&amp; \\hfill 3\\\\ \\hfill 0&amp; \\hfill 1\\end{array}\\right]-\\left[\\begin{array}{rr}\\hfill 8&amp; \\hfill 1\\\\ \\hfill 5&amp; \\hfill 4\\end{array}\\right]\\hfill \\\\ &amp; =\\left[\\begin{array}{rrr}\\hfill -2 - 8&amp; \\hfill &amp; \\hfill 3 - 1\\\\ \\hfill 0 - 5&amp; \\hfill &amp; \\hfill 1 - 4\\end{array}\\right]\\hfill \\\\ &amp; =\\left[\\begin{array}{rrr}\\hfill -10&amp; \\hfill &amp; \\hfill 2\\\\ \\hfill -5&amp; \\hfill &amp; \\hfill -3\\end{array}\\right]\\hfill \\end{array}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox examples\">\n<h3>tip for success<\/h3>\nBe sure to handle negative numbers carefully when finding the difference of two matrices. In the example using the 3 X 3 matrix below, for instance, subtracting [latex]a_{13} - b_{13}[\/latex] results in [latex]-2 - (-2) = 0[\/latex]\n\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Sum and Difference of Two 3 x 3 Matrices<\/h3>\nGiven [latex]A[\/latex] and [latex]B:[\/latex]\n<ol>\n \t<li>Find the sum.<\/li>\n \t<li>Find the difference.<\/li>\n<\/ol>\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{rrr}\\hfill 2&amp; \\hfill -10&amp; \\hfill -2\\\\ \\hfill 14&amp; \\hfill 12&amp; \\hfill 10\\\\ \\hfill 4&amp; \\hfill -2&amp; \\hfill 2\\end{array}\\right]\\text{ and }B=\\left[\\begin{array}{rrr}\\hfill 6&amp; \\hfill 10&amp; \\hfill -2\\\\ \\hfill 0&amp; \\hfill -12&amp; \\hfill -4\\\\ \\hfill -5&amp; \\hfill 2&amp; \\hfill -2\\end{array}\\right][\/latex]<\/p>\n[reveal-answer q=\"286263\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"286263\"]\n<ol>\n \t<li>Add the corresponding entries.\n<div style=\"text-align: center;\">[latex]\\begin{array}{ll}\\hfill \\\\ A+B&amp; =\\left[\\begin{array}{rrr}\\hfill 2&amp; \\hfill -10&amp; \\hfill -2\\\\ \\hfill 14&amp; \\hfill 12&amp; \\hfill 10\\\\ \\hfill 4&amp; \\hfill -2&amp; \\hfill 2\\end{array}\\right]+\\left[\\begin{array}{rrr}\\hfill 6&amp; \\hfill 10&amp; \\hfill -2\\\\ \\hfill 0&amp; \\hfill -12&amp; \\hfill -4\\\\ \\hfill -5&amp; \\hfill 2&amp; \\hfill -2\\end{array}\\right]\\hfill \\\\ &amp; =\\left[\\begin{array}{rrr}\\hfill 2+6&amp; \\hfill -10+10&amp; \\hfill -2 - 2\\\\ \\hfill 14+0&amp; \\hfill 12 - 12&amp; \\hfill 10 - 4\\\\ \\hfill 4 - 5&amp; \\hfill -2+2&amp; \\hfill 2 - 2\\end{array}\\right]\\hfill \\\\ &amp; =\\left[\\begin{array}{rrr}\\hfill 8&amp; \\hfill 0&amp; \\hfill -4\\\\ \\hfill 14&amp; \\hfill 0&amp; \\hfill 6\\\\ \\hfill -1&amp; \\hfill 0&amp; \\hfill 0\\end{array}\\right]\\hfill \\end{array}[\/latex]<\/div><\/li>\n \t<li>Subtract the corresponding entries.\n<div style=\"text-align: center;\">[latex]\\begin{array}{ll}\\hfill \\\\ A-B &amp; =\\left[\\begin{array}{rrr}\\hfill 2&amp; \\hfill -10&amp; \\hfill -2\\\\ \\hfill 14&amp; \\hfill 12&amp; \\hfill 10\\\\ \\hfill 4&amp; \\hfill -2&amp; \\hfill 2\\end{array}\\right]-\\left[\\begin{array}{rrr}\\hfill 6&amp; \\hfill 10&amp; \\hfill -2\\\\ \\hfill 0&amp; \\hfill -12&amp; \\hfill -4\\\\ \\hfill -5&amp; \\hfill 2&amp; \\hfill -2\\end{array}\\right]\\hfill \\\\ &amp; =\\left[\\begin{array}{rrr}\\hfill 2 - 6&amp; \\hfill -10 - 10&amp; \\hfill -2+2\\\\ \\hfill 14 - 0&amp; \\hfill 12+12&amp; \\hfill 10+4\\\\ \\hfill 4+5&amp; \\hfill -2 - 2&amp; \\hfill 2+2\\end{array}\\right]\\hfill \\\\ &amp; =\\left[\\begin{array}{rrr}\\hfill -4&amp; \\hfill -20&amp; \\hfill 0\\\\ \\hfill 14&amp; \\hfill 24&amp; \\hfill 14\\\\ \\hfill 9&amp; \\hfill -4&amp; \\hfill 4\\end{array}\\right]\\hfill \\end{array}[\/latex]<\/div><\/li>\n<\/ol>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nAdd matrix [latex]A[\/latex] and matrix [latex]B[\/latex].\n\n[latex]A=\\left[\\begin{array}{rr}\\hfill 2&amp; \\hfill 6\\\\ \\hfill 1&amp; \\hfill 0\\\\ \\hfill 1&amp; \\hfill -3\\end{array}\\right]\\text{ and }B=\\left[\\begin{array}{rr}\\hfill 3&amp; \\hfill -2\\\\ \\hfill 1&amp; \\hfill 5\\\\ \\hfill -4&amp; \\hfill 3\\end{array}\\right][\/latex]\n\n[reveal-answer q=\"644182\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"644182\"]\n\n[latex]A+B=\\left[\\begin{array}{c}2\\\\ 1\\\\ 1\\end{array}\\begin{array}{c}\\text{ }\\text{ }\\text{ }6\\\\ \\text{ }\\text{ }\\text{ }0\\\\ -3\\end{array}\\right]+\\left[\\begin{array}{c}3\\\\ 1\\\\ -4\\end{array}\\begin{array}{c}-2\\\\ 5\\\\ 3\\end{array}\\right]=\\left[\\begin{array}{c}2+3\\\\ 1+1\\\\ 1+\\left(-4\\right)\\end{array}\\begin{array}{c}6+\\left(-2\\right)\\\\ 0+5\\\\ -3+3\\end{array}\\right]=\\left[\\begin{array}{c}5\\\\ 2\\\\ -3\\end{array}\\begin{array}{c}4\\\\ 5\\\\ 0\\end{array}\\right][\/latex][\/hidden-answer]\n\n[ohm_question]6390[\/ohm_question]\n\n<\/div>\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Determine the dimensions of a matrix.<\/li>\n<li>Add and subtract two matrices.<\/li>\n<\/ul>\n<\/div>\n<p>To solve a problem like the one described for the soccer teams, we can use a <strong>matrix<\/strong>, which is a rectangular array of numbers. A <strong>row<\/strong> in a matrix is a set of numbers that are aligned horizontally. A <strong>column<\/strong> in a matrix is a set of numbers that are aligned vertically. Each number is an <strong>entry<\/strong>, sometimes called an element, of the matrix. Matrices (plural) are enclosed in [ ] or ( ) and are usually named with capital letters. For example, three matrices named [latex]A,B,\\text{}[\/latex] and [latex]C[\/latex] are shown below.<\/p>\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{cc}1& 2\\\\ 3& 4\\end{array}\\right],B=\\left[\\begin{array}{ccc}\\hfill 1& \\hfill 2& \\hfill 7\\\\ \\hfill 0& \\hfill -5& \\hfill 6\\\\ \\hfill 7& \\hfill 8& \\hfill 2\\end{array}\\right],C=\\left[\\begin{array}{c}\\hfill -1\\\\ \\hfill 0\\\\ \\hfill 3\\end{array}\\begin{array}{c}3\\\\ 2\\\\ 1\\end{array}\\right][\/latex]<\/p>\n<h2>Describing Matrices<\/h2>\n<p>A matrix is often referred to by its size or dimensions: [latex]\\text{ }m\\text{ }\\times \\text{ }n\\text{ }[\/latex] indicating [latex]m[\/latex] rows and [latex]n[\/latex] columns. Matrix entries are defined first by row and then by column. For example, to locate the entry in matrix [latex]A[\/latex] identified as [latex]{a}_{ij},\\text{}[\/latex] we look for the entry in row [latex]i,\\text{}[\/latex] column [latex]j[\/latex]. In matrix [latex]A[\/latex] shown below, the entry in row 2, column 3 is [latex]{a}_{23}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\\\\ {a}_{21}& {a}_{22}& {a}_{23}\\\\ {a}_{31}& {a}_{32}& {a}_{33}\\end{array}\\right][\/latex]<\/p>\n<div class=\"textbox examples\">\n<h3>how are those subscripts pronounced?<\/h3>\n<p>Since the subscripts in a matrix refer to row and column entry locations, the numbers are pronounced distinctly from one another. To refer to the 1st row, third column entry in the matrix [latex]A[\/latex] above, we say &#8220;[latex]a \\text{ one }\\text{three}[\/latex].&#8221;<\/p>\n<\/div>\n<p>A <strong>square matrix<\/strong> is a matrix with dimensions [latex]\\text{ }n\\text{ }\\times \\text{ }n,\\text{}[\/latex] meaning that it has the same number of rows as columns. The [latex]3\\times 3[\/latex] matrix above is an example of a square matrix.<\/p>\n<p>A <strong>row matrix<\/strong> is a matrix consisting of one row with dimensions [latex]1\\text{ }\\times \\text{ }n[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\left[\\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\\end{array}\\right][\/latex]<\/p>\n<p>A <strong>column matrix<\/strong> is a matrix consisting of one column with dimensions [latex]m\\text{ }\\times \\text{ }1[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\left[\\begin{array}{c}{a}_{11}\\\\ {a}_{21}\\\\ {a}_{31}\\end{array}\\right][\/latex]<\/p>\n<p>A matrix may be used to represent a system of equations. In these cases, the numbers represent the coefficients of the variables in the system. Matrices often make solving systems of equations easier because they are not encumbered with variables. We will investigate this idea further in the next section, but first we will look at basic <strong>matrix operations<\/strong>.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Matrices<\/h3>\n<p>A <strong>matrix<\/strong> is a rectangular array of numbers that is usually named by a capital letter: [latex]A,B,C,\\text{}[\/latex] and so on. Each entry in a matrix is referred to as [latex]{a}_{ij}[\/latex], such that [latex]i[\/latex] represents the row and [latex]j[\/latex] represents the column. Matrices are often referred to by their dimensions: [latex]m\\times n[\/latex] indicating [latex]m[\/latex] rows and [latex]n[\/latex] columns.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Dimensions of the Given Matrix and Locating Entries<\/h3>\n<p>Given matrix [latex]A:[\/latex]<\/p>\n<ol>\n<li>What are the dimensions of matrix [latex]A?[\/latex]<\/li>\n<li>What are the entries at [latex]{a}_{31}[\/latex] and [latex]{a}_{22}?[\/latex]<\/li>\n<\/ol>\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{rrrr}\\hfill 2& \\hfill & \\hfill 1& \\hfill 0\\\\ \\hfill 2& \\hfill & \\hfill 4& \\hfill 7\\\\ \\hfill 3& \\hfill & \\hfill 1& \\hfill -2\\end{array}\\right][\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q832047\">Show Solution<\/span><\/p>\n<div id=\"q832047\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>The dimensions are [latex]\\text{ }3\\text{ }\\times \\text{ }3\\text{ }[\/latex] because there are three rows and three columns.<\/li>\n<li>Entry [latex]{a}_{31}[\/latex] is the number at row 3, column 1 which is 3. The entry [latex]{a}_{22}[\/latex] is the number at row 2, column 2 which is 4. Remember, the row comes first, then the column.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm6388\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=6388&theme=oea&iframe_resize_id=ohm6388&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Adding and Subtracting Matrices<\/h2>\n<p>We use matrices to list data or to represent systems. Because the entries are numbers, we can perform operations on matrices. We add or subtract matrices by adding or subtracting corresponding entries.<\/p>\n<p>In order to do this, the entries must correspond. Therefore, <em>addition and subtraction of matrices is only possible when the matrices have the same dimensions<\/em>. We can add or subtract a [latex]\\text{ }3\\text{ }\\times \\text{ }3\\text{ }[\/latex] matrix and another [latex]\\text{ }3\\text{ }\\times \\text{ }3\\text{ }[\/latex] matrix, but we cannot add or subtract a [latex]\\text{ }2\\text{ }\\times \\text{ }3\\text{ }[\/latex] matrix and a [latex]\\text{ }3\\text{ }\\times \\text{ }3\\text{ }[\/latex] matrix because some entries in one matrix will not have a corresponding entry in the other matrix.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Adding and Subtracting Matrices<\/h3>\n<p>Given matrices [latex]A[\/latex] and [latex]B[\/latex] of like dimensions, addition and subtraction of [latex]A[\/latex] and [latex]B[\/latex] will produce matrix [latex]C[\/latex] or&nbsp;matrix [latex]D[\/latex] of the same dimension.<\/p>\n<p style=\"text-align: center;\">[latex]A+B=C\\text{ such that }{a}_{ij}+{b}_{ij}={c}_{ij}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]A-B=D\\text{ such that }{a}_{ij}-{b}_{ij}={d}_{ij}[\/latex]<\/p>\n<p>Matrix addition is commutative.<\/p>\n<p style=\"text-align: center;\">[latex]A+B=B+A[\/latex]<\/p>\n<p>It is also associative.<\/p>\n<p style=\"text-align: center;\">[latex]\\left(A+B\\right)+C=A+\\left(B+C\\right)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Example: Finding the Sum of Matrices<\/h3>\n<p>Find the sum of [latex]A[\/latex] and [latex]B \\text{}[\/latex] given<\/p>\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{cc}a& b\\\\ c& d\\end{array}\\right]\\text{ and }B=\\left[\\begin{array}{cc}e& f\\\\ g& h\\end{array}\\right][\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q3634\">Show Solution<\/span><\/p>\n<div id=\"q3634\" class=\"hidden-answer\" style=\"display: none\">\n<p>Add corresponding entries.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}A+B & =\\left[\\begin{array}{cc}a& b\\\\ c& d\\end{array}\\right]+\\left[\\begin{array}{cc}e& f\\\\ g& h\\end{array}\\right]\\hfill \\\\ & =\\left[\\begin{array}{ccc}a+e& & b+f\\\\ c+g& & d+h\\end{array}\\right]\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm1079\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=1079&theme=oea&iframe_resize_id=ohm1079&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Adding Matrix <em>A <\/em>and Matrix B<\/h3>\n<p>Find the sum of [latex]A[\/latex] and [latex]B[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{cc}4& 1\\\\ 3& 2\\end{array}\\right]\\text{ and }B=\\left[\\begin{array}{cc}5& 9\\\\ 0& 7\\end{array}\\right][\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q512431\">Show Solution<\/span><\/p>\n<div id=\"q512431\" class=\"hidden-answer\" style=\"display: none\">\n<p>Add corresponding entries. Add the entry in row 1, column 1, [latex]{a}_{11},\\text{}[\/latex] of matrix [latex]A[\/latex] to the entry in row 1, column 1, [latex]{b}_{11}[\/latex], of [latex]B[\/latex]. Continue the pattern until all entries have been added.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll}A+B & =\\left[\\begin{array}{cc}4& 1\\\\ 3& 2\\end{array}\\right]+\\left[\\begin{array}{cc}5& 9\\\\ 0& 7\\end{array}\\right]\\hfill \\\\ & =\\left[\\begin{array}{ccc}4+5& & 1+9\\\\ 3+0& & 2+7\\end{array}\\right]\\hfill \\\\ & =\\left[\\begin{array}{cc}9& 10\\\\ 3& 9\\end{array}\\right]\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Difference of Two Matrices<\/h3>\n<p>Find the difference of [latex]A[\/latex] and [latex]B[\/latex].<br \/>\n[latex]A=\\left[\\begin{array}{cc}\\hfill -2& \\hfill 3\\\\ \\hfill 0& \\hfill 1\\end{array}\\right]\\text{ and }B=\\left[\\begin{array}{cc}8& 1\\\\ 5& 4\\end{array}\\right][\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q83802\">Show Solution<\/span><\/p>\n<div id=\"q83802\" class=\"hidden-answer\" style=\"display: none\">\n<p>We subtract the corresponding entries of each matrix.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll}A-B & =\\left[\\begin{array}{rr}\\hfill -2& \\hfill 3\\\\ \\hfill 0& \\hfill 1\\end{array}\\right]-\\left[\\begin{array}{rr}\\hfill 8& \\hfill 1\\\\ \\hfill 5& \\hfill 4\\end{array}\\right]\\hfill \\\\ & =\\left[\\begin{array}{rrr}\\hfill -2 - 8& \\hfill & \\hfill 3 - 1\\\\ \\hfill 0 - 5& \\hfill & \\hfill 1 - 4\\end{array}\\right]\\hfill \\\\ & =\\left[\\begin{array}{rrr}\\hfill -10& \\hfill & \\hfill 2\\\\ \\hfill -5& \\hfill & \\hfill -3\\end{array}\\right]\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>tip for success<\/h3>\n<p>Be sure to handle negative numbers carefully when finding the difference of two matrices. In the example using the 3 X 3 matrix below, for instance, subtracting [latex]a_{13} - b_{13}[\/latex] results in [latex]-2 - (-2) = 0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Sum and Difference of Two 3 x 3 Matrices<\/h3>\n<p>Given [latex]A[\/latex] and [latex]B:[\/latex]<\/p>\n<ol>\n<li>Find the sum.<\/li>\n<li>Find the difference.<\/li>\n<\/ol>\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{rrr}\\hfill 2& \\hfill -10& \\hfill -2\\\\ \\hfill 14& \\hfill 12& \\hfill 10\\\\ \\hfill 4& \\hfill -2& \\hfill 2\\end{array}\\right]\\text{ and }B=\\left[\\begin{array}{rrr}\\hfill 6& \\hfill 10& \\hfill -2\\\\ \\hfill 0& \\hfill -12& \\hfill -4\\\\ \\hfill -5& \\hfill 2& \\hfill -2\\end{array}\\right][\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q286263\">Show Solution<\/span><\/p>\n<div id=\"q286263\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>Add the corresponding entries.\n<div style=\"text-align: center;\">[latex]\\begin{array}{ll}\\hfill \\\\ A+B& =\\left[\\begin{array}{rrr}\\hfill 2& \\hfill -10& \\hfill -2\\\\ \\hfill 14& \\hfill 12& \\hfill 10\\\\ \\hfill 4& \\hfill -2& \\hfill 2\\end{array}\\right]+\\left[\\begin{array}{rrr}\\hfill 6& \\hfill 10& \\hfill -2\\\\ \\hfill 0& \\hfill -12& \\hfill -4\\\\ \\hfill -5& \\hfill 2& \\hfill -2\\end{array}\\right]\\hfill \\\\ & =\\left[\\begin{array}{rrr}\\hfill 2+6& \\hfill -10+10& \\hfill -2 - 2\\\\ \\hfill 14+0& \\hfill 12 - 12& \\hfill 10 - 4\\\\ \\hfill 4 - 5& \\hfill -2+2& \\hfill 2 - 2\\end{array}\\right]\\hfill \\\\ & =\\left[\\begin{array}{rrr}\\hfill 8& \\hfill 0& \\hfill -4\\\\ \\hfill 14& \\hfill 0& \\hfill 6\\\\ \\hfill -1& \\hfill 0& \\hfill 0\\end{array}\\right]\\hfill \\end{array}[\/latex]<\/div>\n<\/li>\n<li>Subtract the corresponding entries.\n<div style=\"text-align: center;\">[latex]\\begin{array}{ll}\\hfill \\\\ A-B & =\\left[\\begin{array}{rrr}\\hfill 2& \\hfill -10& \\hfill -2\\\\ \\hfill 14& \\hfill 12& \\hfill 10\\\\ \\hfill 4& \\hfill -2& \\hfill 2\\end{array}\\right]-\\left[\\begin{array}{rrr}\\hfill 6& \\hfill 10& \\hfill -2\\\\ \\hfill 0& \\hfill -12& \\hfill -4\\\\ \\hfill -5& \\hfill 2& \\hfill -2\\end{array}\\right]\\hfill \\\\ & =\\left[\\begin{array}{rrr}\\hfill 2 - 6& \\hfill -10 - 10& \\hfill -2+2\\\\ \\hfill 14 - 0& \\hfill 12+12& \\hfill 10+4\\\\ \\hfill 4+5& \\hfill -2 - 2& \\hfill 2+2\\end{array}\\right]\\hfill \\\\ & =\\left[\\begin{array}{rrr}\\hfill -4& \\hfill -20& \\hfill 0\\\\ \\hfill 14& \\hfill 24& \\hfill 14\\\\ \\hfill 9& \\hfill -4& \\hfill 4\\end{array}\\right]\\hfill \\end{array}[\/latex]<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Add matrix [latex]A[\/latex] and matrix [latex]B[\/latex].<\/p>\n<p>[latex]A=\\left[\\begin{array}{rr}\\hfill 2& \\hfill 6\\\\ \\hfill 1& \\hfill 0\\\\ \\hfill 1& \\hfill -3\\end{array}\\right]\\text{ and }B=\\left[\\begin{array}{rr}\\hfill 3& \\hfill -2\\\\ \\hfill 1& \\hfill 5\\\\ \\hfill -4& \\hfill 3\\end{array}\\right][\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q644182\">Show Solution<\/span><\/p>\n<div id=\"q644182\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]A+B=\\left[\\begin{array}{c}2\\\\ 1\\\\ 1\\end{array}\\begin{array}{c}\\text{ }\\text{ }\\text{ }6\\\\ \\text{ }\\text{ }\\text{ }0\\\\ -3\\end{array}\\right]+\\left[\\begin{array}{c}3\\\\ 1\\\\ -4\\end{array}\\begin{array}{c}-2\\\\ 5\\\\ 3\\end{array}\\right]=\\left[\\begin{array}{c}2+3\\\\ 1+1\\\\ 1+\\left(-4\\right)\\end{array}\\begin{array}{c}6+\\left(-2\\right)\\\\ 0+5\\\\ -3+3\\end{array}\\right]=\\left[\\begin{array}{c}5\\\\ 2\\\\ -3\\end{array}\\begin{array}{c}4\\\\ 5\\\\ 0\\end{array}\\right][\/latex]<\/p><\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm6390\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=6390&theme=oea&iframe_resize_id=ohm6390&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\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-401\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <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 class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>Question ID 6388. <strong>Authored by<\/strong>: Dow,David D.. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><\/ul><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":6,"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\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\"},{\"type\":\"cc\",\"description\":\"Question ID 6388\",\"author\":\"Dow,David D.\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"}]","CANDELA_OUTCOMES_GUID":"2fb87fbe-c67c-4068-9b0e-ae95eec362d2","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-401","chapter","type-chapter","status-publish","hentry"],"part":395,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/pressbooks\/v2\/chapters\/401","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":0,"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/pressbooks\/v2\/chapters\/401\/revisions"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/pressbooks\/v2\/parts\/395"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/pressbooks\/v2\/chapters\/401\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/wp\/v2\/media?parent=401"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=401"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/wp\/v2\/contributor?post=401"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/wp\/v2\/license?post=401"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}