{"id":1787,"date":"2015-11-12T18:30:45","date_gmt":"2015-11-12T18:30:45","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=1787"},"modified":"2017-04-03T21:12:56","modified_gmt":"2017-04-03T21:12:56","slug":"finding-the-sum-and-difference-of-two-matrices","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/chapter\/finding-the-sum-and-difference-of-two-matrices\/","title":{"raw":"Finding the Sum and Difference of Two Matrices","rendered":"Finding the Sum and Difference of Two Matrices"},"content":{"raw":"

To solve a problem like the one described for the soccer teams, we can use a matrix<\/strong>, which is a rectangular array of numbers. A row<\/strong> in a matrix is a set of numbers that are aligned horizontally. A column<\/strong> in a matrix is a set of numbers that are aligned vertically. Each number is an 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.\r\n<\/p>

[latex]A=\\left[\\begin{array}{cc}1& 2\\\\ 3& 4\\end{array}\\right],B=\\left[\\begin{array}{ccc}1& 2& 7\\\\ 0& -5& 6\\\\ 7& 8& 2\\end{array}\\right],C=\\left[\\begin{array}{c}-1\\\\ 0\\\\ 3\\end{array}\\begin{array}{c}3\\\\ 2\\\\ 1\\end{array}\\right][\/latex]<\/div>\r\n

Describing Matrices<\/h2>\r\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\\text{,} \\hspace{0.17em}[\/latex] shown below, the entry in row 2, column 3 is [latex]{a}_{23}[\/latex].\r\n
[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]<\/div>\r\nA 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.\r\n\r\nA row matrix<\/strong> is a matrix consisting of one row with dimensions [latex]1\\text{ }\\times \\text{ }n[\/latex].\r\n
[latex]\\left[\\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\\end{array}\\right][\/latex]<\/div>\r\nA column matrix<\/strong> is a matrix consisting of one column with dimensions [latex]m\\text{ }\\times \\text{ }1[\/latex].\r\n
[latex]\\left[\\begin{array}{c}{a}_{11}\\\\ {a}_{21}\\\\ {a}_{31}\\end{array}\\right][\/latex]<\/div>\r\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 matrix operations<\/strong>.\r\n
\r\n

A General Note: Matrices<\/h3>\r\nA 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.\r\n\r\n<\/div>\r\n
\r\n

Example 1: Finding the Dimensions of the Given Matrix and Locating Entries<\/h3>\r\nGiven matrix [latex]A:[\/latex]\r\n
  1. What are the dimensions of matrix [latex]A?[\/latex]<\/li>\r\n\t
  2. What are the entries at [latex]{a}_{31}[\/latex] and [latex]{a}_{22}?[\/latex]\r\n
    [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]<\/div><\/li>\r\n<\/ol><\/div>\r\n
    \r\n

    Solution<\/h3>\r\n
    1. The dimensions are [latex]\\text{ }3\\text{ }\\times \\text{ }3\\text{ }[\/latex] because there are three rows and three columns.<\/li>\r\n\t
    2. 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>\r\n<\/ol><\/div>\r\n

      Adding and Subtracting Matrices<\/h2>\r\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.\r\n\r\nIn order to do this, the entries must correspond. Therefore, 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.\r\n
      \r\n

      A General Note: Adding and Subtracting Matrices<\/h3>\r\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\u00a0matrix [latex]D[\/latex] of the same dimension.\r\n
      [latex]A+B=C\\text{ such that }{a}_{ij}+{b}_{ij}={c}_{ij}[\/latex]<\/div>\r\n
      [latex]A-B=D\\text{ such that }{a}_{ij}-{b}_{ij}={d}_{ij}[\/latex]<\/div>\r\nMatrix addition is commutative.\r\n
      [latex]A+B=B+A[\/latex]<\/div>\r\nIt is also associative.\r\n
      [latex]\\left(A+B\\right)+C=A+\\left(B+C\\right)[\/latex]<\/div>\r\n<\/div>\r\n
      \r\n

      Example 2: Finding the Sum of Matrices<\/h3>\r\nFind the sum of [latex]A[\/latex] and [latex]B,\\text{}[\/latex] given\r\n
      [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]<\/div>\r\n<\/div>\r\n
      \r\n

      Solution<\/h3>\r\nAdd corresponding entries.\r\n
      [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 \\\\ \\text{ }=\\left[\\begin{array}{ccc}a+e& & b+f\\\\ c+g& & d+h\\end{array}\\right]\\hfill \\end{array}[\/latex]<\/div>\r\n<\/div>\r\n
      \r\n

      Example 3: Adding Matrix A <\/em>and Matrix <>B<\/h3>\r\nFind the sum of [latex]A[\/latex] and [latex]B[\/latex].\r\n
      [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]<\/div>\r\n<\/div>\r\n
      \r\n

      Solution<\/h3>\r\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.\r\n
      [latex]\\begin{array}{l}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 \\\\ \\text{ }=\\left[\\begin{array}{ccc}4+5& & 1+9\\\\ 3+0& & 2+7\\end{array}\\right]\\hfill \\\\ \\text{ }=\\left[\\begin{array}{cc}9& 10\\\\ 3& 9\\end{array}\\right]\\hfill \\end{array}[\/latex]<\/div>\r\n<\/div>\r\n
      \r\n

      Example 4: Finding the Difference of Two Matrices<\/h3>\r\nFind the difference of [latex]A[\/latex] and [latex]B[\/latex].\r\n
      [latex]A=\\left[\\begin{array}{cc}-2& 3\\\\ 0& 1\\end{array}\\right]\\text{ and }B=\\left[\\begin{array}{cc}8& 1\\\\ 5& 4\\end{array}\\right][\/latex]<\/div>\r\n<\/div>\r\n
      \r\n

      Solution<\/h3>\r\nWe subtract the corresponding entries of each matrix.\r\n
      [latex]\\begin{array}{l}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 \\\\ \\text{ }=\\left[\\begin{array}{rrr}\\hfill -2 - 8& \\hfill & \\hfill 3 - 1\\\\ \\hfill 0 - 5& \\hfill & \\hfill 1 - 4\\end{array}\\right]\\hfill \\\\ \\text{ }=\\left[\\begin{array}{rrr}\\hfill -10& \\hfill & \\hfill 2\\\\ \\hfill -5& \\hfill & \\hfill -3\\end{array}\\right]\\hfill \\end{array}[\/latex]<\/div>\r\n<\/div>\r\n
      \r\n

      Example 5: Finding the Sum and Difference of Two 3 x 3 Matrices<\/h3>\r\nGiven [latex]A[\/latex] and [latex]B:[\/latex]\r\n
      1. Find the sum.<\/li>\r\n\t
      2. Find the difference.<\/li>\r\n<\/ol>
        [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]<\/div>\r\n<\/div>\r\n
        \r\n

        Solution<\/h3>\r\n
        1. Add the corresponding entries.\r\n
          [latex]\\begin{array}{l}\\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><\/li>\r\n\t
        2. Subtract the corresponding entries.\r\n
          [latex]\\begin{array}{l}\\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><\/li>\r\n<\/ol><\/div>\r\n
          \r\n

          Try It 1<\/h3>\r\nAdd matrix [latex]A[\/latex] and matrix [latex]B[\/latex].\r\n
          [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]<\/div>\r\n
          Solution<\/a><\/div>\r\n<\/div>","rendered":"

          To solve a problem like the one described for the soccer teams, we can use a matrix<\/strong>, which is a rectangular array of numbers. A row<\/strong> in a matrix is a set of numbers that are aligned horizontally. A column<\/strong> in a matrix is a set of numbers that are aligned vertically. Each number is an 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>\n

          [latex]A=\\left[\\begin{array}{cc}1& 2\\\\ 3& 4\\end{array}\\right],B=\\left[\\begin{array}{ccc}1& 2& 7\\\\ 0& -5& 6\\\\ 7& 8& 2\\end{array}\\right],C=\\left[\\begin{array}{c}-1\\\\ 0\\\\ 3\\end{array}\\begin{array}{c}3\\\\ 2\\\\ 1\\end{array}\\right][\/latex]<\/div>\n

          Describing Matrices<\/h2>\n

          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\\text{,} \\hspace{0.17em}[\/latex] shown below, the entry in row 2, column 3 is [latex]{a}_{23}[\/latex].<\/p>\n

          [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]<\/div>\n

          A 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

          A row matrix<\/strong> is a matrix consisting of one row with dimensions [latex]1\\text{ }\\times \\text{ }n[\/latex].<\/p>\n

          [latex]\\left[\\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\\end{array}\\right][\/latex]<\/div>\n

          A column matrix<\/strong> is a matrix consisting of one column with dimensions [latex]m\\text{ }\\times \\text{ }1[\/latex].<\/p>\n

          [latex]\\left[\\begin{array}{c}{a}_{11}\\\\ {a}_{21}\\\\ {a}_{31}\\end{array}\\right][\/latex]<\/div>\n

          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 matrix operations<\/strong>.<\/p>\n

          \n

          A General Note: Matrices<\/h3>\n

          A 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

          \n

          Example 1: Finding the Dimensions of the Given Matrix and Locating Entries<\/h3>\n

          Given matrix [latex]A:[\/latex]<\/p>\n

            \n
          1. What are the dimensions of matrix [latex]A?[\/latex]<\/li>\n
          2. What are the entries at [latex]{a}_{31}[\/latex] and [latex]{a}_{22}?[\/latex]\n
            [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]<\/div>\n<\/li>\n<\/ol>\n<\/div>\n
            \n

            Solution<\/h3>\n
              \n
            1. The dimensions are [latex]\\text{ }3\\text{ }\\times \\text{ }3\\text{ }[\/latex] because there are three rows and three columns.<\/li>\n
            2. 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

              Adding and Subtracting Matrices<\/h2>\n

              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

              In order to do this, the entries must correspond. Therefore, 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

              \n

              A General Note: Adding and Subtracting Matrices<\/h3>\n

              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\u00a0matrix [latex]D[\/latex] of the same dimension.<\/p>\n

              [latex]A+B=C\\text{ such that }{a}_{ij}+{b}_{ij}={c}_{ij}[\/latex]<\/div>\n
              [latex]A-B=D\\text{ such that }{a}_{ij}-{b}_{ij}={d}_{ij}[\/latex]<\/div>\n

              Matrix addition is commutative.<\/p>\n

              [latex]A+B=B+A[\/latex]<\/div>\n

              It is also associative.<\/p>\n

              [latex]\\left(A+B\\right)+C=A+\\left(B+C\\right)[\/latex]<\/div>\n<\/div>\n
              \n

              Example 2: Finding the Sum of Matrices<\/h3>\n

              Find the sum of [latex]A[\/latex] and [latex]B,\\text{}[\/latex] given<\/p>\n

              [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]<\/div>\n<\/div>\n
              \n

              Solution<\/h3>\n

              Add corresponding entries.<\/p>\n

              [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 \\\\ \\text{ }=\\left[\\begin{array}{ccc}a+e& & b+f\\\\ c+g& & d+h\\end{array}\\right]\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n
              \n

              Example 3: Adding Matrix A <\/em>and Matrix <>B<\/h3>\n

              Find the sum of [latex]A[\/latex] and [latex]B[\/latex].<\/p>\n

              [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]<\/div>\n<\/div>\n
              \n

              Solution<\/h3>\n

              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

              [latex]\\begin{array}{l}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 \\\\ \\text{ }=\\left[\\begin{array}{ccc}4+5& & 1+9\\\\ 3+0& & 2+7\\end{array}\\right]\\hfill \\\\ \\text{ }=\\left[\\begin{array}{cc}9& 10\\\\ 3& 9\\end{array}\\right]\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n
              \n

              Example 4: Finding the Difference of Two Matrices<\/h3>\n

              Find the difference of [latex]A[\/latex] and [latex]B[\/latex].<\/p>\n

              [latex]A=\\left[\\begin{array}{cc}-2& 3\\\\ 0& 1\\end{array}\\right]\\text{ and }B=\\left[\\begin{array}{cc}8& 1\\\\ 5& 4\\end{array}\\right][\/latex]<\/div>\n<\/div>\n
              \n

              Solution<\/h3>\n

              We subtract the corresponding entries of each matrix.<\/p>\n

              [latex]\\begin{array}{l}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 \\\\ \\text{ }=\\left[\\begin{array}{rrr}\\hfill -2 - 8& \\hfill & \\hfill 3 - 1\\\\ \\hfill 0 - 5& \\hfill & \\hfill 1 - 4\\end{array}\\right]\\hfill \\\\ \\text{ }=\\left[\\begin{array}{rrr}\\hfill -10& \\hfill & \\hfill 2\\\\ \\hfill -5& \\hfill & \\hfill -3\\end{array}\\right]\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n
              \n

              Example 5: Finding the Sum and Difference of Two 3 x 3 Matrices<\/h3>\n

              Given [latex]A[\/latex] and [latex]B:[\/latex]<\/p>\n

                \n
              1. Find the sum.<\/li>\n
              2. Find the difference.<\/li>\n<\/ol>\n
                [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]<\/div>\n<\/div>\n
                \n

                Solution<\/h3>\n
                  \n
                1. Add the corresponding entries.\n
                  [latex]\\begin{array}{l}\\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
                2. Subtract the corresponding entries.\n
                  [latex]\\begin{array}{l}\\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
                  \n

                  Try It 1<\/h3>\n

                  Add matrix [latex]A[\/latex] and matrix [latex]B[\/latex].<\/p>\n

                  [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]<\/div>\n
                  Solution<\/a><\/div>\n<\/div>\n\n\t\t\t
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                  Candela Citations<\/h3>\n\t\t\t\t\t
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