{"id":1377,"date":"2025-07-24T19:14:32","date_gmt":"2025-07-24T19:14:32","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1377"},"modified":"2026-03-24T17:49:26","modified_gmt":"2026-03-24T17:49:26","slug":"solving-systems-with-inverses-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/solving-systems-with-inverses-learn-it-3\/","title":{"raw":"Solving Systems with Inverses: Learn It 3","rendered":"Solving Systems with Inverses: Learn It 3"},"content":{"raw":"

Finding the Multiplicative Inverse of 2\u00d72 Matrices Using a Formula<\/h2>\r\nWhen we need to find the multiplicative inverse<\/strong> of a [latex]2\\times 2[\/latex] matrix, we can use a special formula instead of using matrix multiplication or augmenting with the identity.\r\n\r\n
\r\n

multiplicative inverse of 2\u00d72 matrices formula<\/h3>\r\nIf [latex]A[\/latex] is a [latex]2\\times 2[\/latex] matrix, such as\r\n

[latex]A=\\left[\\begin{array}{rrr}\\hfill a& \\hfill & \\hfill b\\\\ \\hfill c& \\hfill & \\hfill d\\end{array}\\right][\/latex]<\/p>\r\nthe multiplicative inverse of [latex]A[\/latex] is given by the formula\r\n

[latex]{A}^{-1}=\\frac{1}{ad-bc}\\left[\\begin{array}{rrr}\\hfill d& \\hfill & \\hfill -b\\\\ \\hfill -c& \\hfill & \\hfill a\\end{array}\\right][\/latex]<\/p>\r\nwhere [latex]ad-bc\\ne 0[\/latex].\r\n\r\n \r\n\r\nIf [latex]ad-bc=0[\/latex], then [latex]A[\/latex] has no inverse.\r\n\r\n<\/section>

Use the formula to find the multiplicative inverse of\r\n

[latex]A=\\left[\\begin{array}{cc}1& -2\\\\ 2& -3\\end{array}\\right][\/latex]<\/p>\r\n[reveal-answer q=\"11636\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"11636\"]\r\n\r\nUsing the formula, we have\r\n

[latex]\\begin{array}{l}{A}^{-1} & =\\frac{1}{\\left(1\\right)\\left(-3\\right)-\\left(-2\\right)\\left(2\\right)}\\left[\\begin{array}{cc}-3& 2\\\\ -2& 1\\end{array}\\right]\\hfill \\\\ & =\\frac{1}{-3+4}\\left[\\begin{array}{cc}-3& 2\\\\ -2& 1\\end{array}\\right]\\hfill \\\\ & =\\left[\\begin{array}{cc}-3& 2\\\\ -2& 1\\end{array}\\right]\\hfill \\end{array}[\/latex]<\/p>\r\nAnalysis of the Solution<\/strong>\r\n\r\nWe can check that our formula works by using one of the other methods to calculate the inverse. Let\u2019s augment [latex]A[\/latex] with the identity.\r\n

[latex]\\left[\\begin{array}{cc|cc}\\hfill 1& \\hfill -2& \\hfill 1& \\hfill 0\\\\ \\hfill 2& \\hfill -3& \\hfill 0& \\hfill 1\\\\ \\end{array}\\right][\/latex]<\/p>\r\nPerform row operations<\/strong> with the goal of turning [latex]A[\/latex] into the identity.\r\n

    \r\n \t
  1. Multiply row 1 by [latex]-2[\/latex] and add to row 2.\r\n
    [latex]\\left[\\begin{array}{cc|cc}\\hfill 1& \\hfill -2& \\hfill 1& \\hfill 0\\\\ \\hfill 0& \\hfill 1& \\hfill -2& \\hfill 1\\\\ \\end{array}\\right][\/latex]<\/center><\/li>\r\n \t
  2. Multiply row 1 by 2 and add to row 1.\r\n
    [latex]\\left[\\begin{array}{cc|cc}\\hfill 1& \\hfill 0& \\hfill -3& \\hfill 2\\\\ \\hfill 0& \\hfill 1& \\hfill -2& \\hfill 1\\\\ \\end{array}\\right][\/latex]<\/center><\/li>\r\n<\/ol>\r\nSo, we have verified our original solution.\r\n

    [latex]{A}^{-1}=\\left[\\begin{array}{cc}-3& 2\\\\ -2& 1\\end{array}\\right][\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>

    [ohm_question hide_question_numbers=1]321755[\/ohm_question]<\/section>
    [ohm_question hide_question_numbers=1]321756[\/ohm_question]<\/section>
    \r\n

    Multiplicative Inverse of 3\u00d73 Matrices<\/h2>\r\nUnfortunately, we do not have a formula similar to the one for a [latex]2\\text{}\\times \\text{}2[\/latex] matrix to find the inverse of a [latex]3\\text{}\\times \\text{}3[\/latex] matrix. But, we can still find the inverse by using a systematic approach involving row operations. This method requires augmenting the given matrix with the identity matrix and performing a series of row operations to transform the original matrix into the identity matrix. The resulting augmented matrix will then have the inverse of the original matrix on its right side.\r\n\r\n
    How To: Given a [latex]3\\times 3[\/latex] matrix, find the inverse<\/strong>\r\n
      \r\n \t
    1. Write the original matrix augmented with the identity matrix on the right.<\/li>\r\n \t
    2. Use elementary row operations so that the identity appears on the left.<\/li>\r\n \t
    3. What is obtained on the right is the inverse of the original matrix.<\/li>\r\n \t
    4. Use matrix multiplication to show that [latex]A{A}^{-1}=I[\/latex] and [latex]{A}^{-1}A=I[\/latex].<\/li>\r\n<\/ol>\r\n<\/section>
      Given the [latex]3\\times 3[\/latex] matrix [latex]A[\/latex], find the inverse.\r\n

      [latex]A=\\left[\\begin{array}{ccc}2& 3& 1\\\\ 3& 3& 1\\\\ 2& 4& 1\\end{array}\\right][\/latex]<\/p>\r\n[reveal-answer q=\"144003\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"144003\"]\r\n\r\nAugment [latex]A[\/latex] with the identity matrix, and then begin row operations until the identity matrix replaces [latex]A[\/latex]. The matrix on the right will be the inverse of [latex]A[\/latex].\r\n

      [latex]\\left[\\begin{array}{ccc|ccc}\\hfill 2& \\hfill 3& \\hfill 1& \\hfill 1& \\hfill 0& \\hfill 0\\\\ \\hfill 3& \\hfill 3& \\hfill 1& \\hfill 0& \\hfill 1& \\hfill 0\\\\ \\hfill 2& \\hfill 4& \\hfill 1& \\hfill 0& \\hfill 0& \\hfill 1\\end{array}\\right]\\stackrel{\\text{Interchange }{R}_{2}\\text{ and }{R}_{1}}{\\to }\\left[\\begin{array}{ccc|ccc}\\hfill 3& \\hfill 3& \\hfill 1& \\hfill 0& \\hfill 1& \\hfill 0\\\\ \\hfill 2& \\hfill 3& \\hfill 1& \\hfill 1& \\hfill 0& \\hfill 0\\\\ \\hfill 2& \\hfill 4& \\hfill 1& \\hfill 0& \\hfill 0& \\hfill 1\\end{array}\\right][\/latex]<\/p>\r\n

      [latex]-{R}_{2}+{R}_{1}={R}_{1}\\to \\left[\\begin{array}{ccc|ccc}\\hfill 1& \\hfill 0& \\hfill 0& \\hfill -1& \\hfill 1& \\hfill 0\\\\ \\hfill 2& \\hfill 3& \\hfill 1& \\hfill 1& \\hfill 0& \\hfill 0\\\\ \\hfill 2& \\hfill 4& \\hfill 1& \\hfill 0& \\hfill 0& \\hfill 1\\end{array}\\right][\/latex]<\/p>\r\n

      [latex]-{R}_{2}+{R}_{3}={R}_{3}\\to \\left[\\begin{array}{ccc|ccc}\\hfill 1& \\hfill 0& \\hfill 0& \\hfill -1& \\hfill 1& \\hfill 0\\\\ \\hfill 2& \\hfill 3& \\hfill 1& \\hfill 1& \\hfill 0& \\hfill 0\\\\ \\hfill 0& \\hfill 1& \\hfill 0& \\hfill -1& \\hfill 0& \\hfill 1\\end{array}\\right][\/latex]<\/p>\r\n

      [latex]{R}_{3}\\leftrightarrow {R}_{2}\\to \\left[\\begin{array}{ccc|ccc}\\hfill 1& \\hfill 0& \\hfill 0& \\hfill -1& \\hfill 1& \\hfill 0\\\\ \\hfill 0& \\hfill 1& \\hfill 0& \\hfill -1& \\hfill 0& \\hfill 1\\\\ \\hfill 2& \\hfill 3& \\hfill 1& \\hfill 1& \\hfill 0& \\hfill 0\\end{array}\\right][\/latex]<\/p>\r\n

      [latex]-2{R}_{1}+{R}_{3}={R}_{3}\\to\\left[\\begin{array}{ccc|ccc}\\hfill 1& \\hfill 0& \\hfill 0& \\hfill -1& \\hfill 1& \\hfill 0\\\\ \\hfill 0& \\hfill 1& \\hfill 0& \\hfill -1& \\hfill 0& \\hfill 1\\\\ \\hfill 0& \\hfill 3& \\hfill 1& \\hfill 3& \\hfill -2& \\hfill 0\\end{array}\\right][\/latex]<\/p>\r\n

      [latex]-3{R}_{2}+{R}_{3}={R}_{3}\\to\\left[\\begin{array}{ccc|ccc}\\hfill 1& \\hfill 0& \\hfill 0& \\hfill -1& \\hfill 1& \\hfill 0\\\\ \\hfill 0& \\hfill 1& \\hfill 0& \\hfill -1& \\hfill 0& \\hfill 1\\\\ \\hfill 0& \\hfill 0& \\hfill 1& \\hfill 6& \\hfill -2& \\hfill -3\\end{array}\\right][\/latex]<\/p>\r\nThus,\r\n

      [latex]{A}^{-1}=B=\\left[\\begin{array}{ccc}-1& 1& 0\\\\ -1& 0& 1\\\\ 6& -2& -3\\end{array}\\right][\/latex]<\/p>\r\nAnalysis of the Solution<\/strong>\r\n\r\nTo prove that [latex]B={A}^{-1}[\/latex], let\u2019s multiply the two matrices together to see if the product equals the identity, if [latex]A{A}^{-1}=I[\/latex] and [latex]{A}^{-1}A=I[\/latex].\r\n

      [latex]\\begin{array}{l}\\begin{array}{l}\\hfill \\\\ \\hfill \\end{array}\\hfill \\\\ A{A}^{-1} & =\\left[\\begin{array}{ccc}2& 3& 1\\\\ 3& 3& 1\\\\ 2& 4& 1\\end{array}\\right]\\text{ }\\left[\\begin{array}{rrr}\\hfill -1& \\hfill 1& \\hfill 0\\\\ \\hfill -1& \\hfill 0& \\hfill 1\\\\ \\hfill 6& \\hfill -2& \\hfill -3\\end{array}\\right]\\hfill \\\\ & =\\left[\\begin{array}{ccc}2\\left(-1\\right)+3\\left(-1\\right)+1\\left(6\\right)& 2\\left(1\\right)+3\\left(0\\right)+1\\left(-2\\right)& 2\\left(0\\right)+3\\left(1\\right)+1\\left(-3\\right)\\\\ 3\\left(-1\\right)+3\\left(-1\\right)+1\\left(6\\right)& 3\\left(1\\right)+3\\left(0\\right)+1\\left(-2\\right)& 3\\left(0\\right)+3\\left(1\\right)+1\\left(-3\\right)\\\\ 2\\left(-1\\right)+4\\left(-1\\right)+1\\left(6\\right)& 2\\left(1\\right)+4\\left(0\\right)+1\\left(-2\\right)& 2\\left(0\\right)+4\\left(1\\right)+1\\left(-3\\right)\\end{array}\\right]\\hfill \\\\ & =\\left[\\begin{array}{ccc}1& 0& 0\\\\ 0& 1& 0\\\\ 0& 0& 1\\end{array}\\right]\\hfill \\end{array}[\/latex]\r\n[latex]\\begin{array}{l}\\begin{array}{l}\\hfill \\\\ \\hfill \\end{array}\\hfill \\\\ {A}^{-1}A & =\\left[\\begin{array}{rrr}\\hfill -1& \\hfill 1& \\hfill 0\\\\ \\hfill -1& \\hfill 0& \\hfill 1\\\\ \\hfill 6& \\hfill -2& \\hfill -3\\end{array}\\right]\\text{ }\\left[\\begin{array}{ccc}2& 3& 1\\\\ 3& 3& 1\\\\ 2& 4& 1\\end{array}\\right]\\hfill \\\\ & =\\left[\\begin{array}{rrr}\\hfill -1\\left(2\\right)+1\\left(3\\right)+0\\left(2\\right)& \\hfill -1\\left(3\\right)+1\\left(3\\right)+0\\left(4\\right)& \\hfill -1\\left(1\\right)+1\\left(1\\right)+0\\left(1\\right)\\\\ \\hfill -1\\left(2\\right)+0\\left(3\\right)+1\\left(2\\right)& \\hfill -1\\left(3\\right)+0\\left(3\\right)+1\\left(4\\right)& \\hfill -1\\left(1\\right)+0\\left(1\\right)+1\\left(1\\right)\\\\ \\hfill 6\\left(2\\right)+-2\\left(3\\right)+-3\\left(2\\right)& \\hfill 6\\left(3\\right)+-2\\left(3\\right)+-3\\left(4\\right)& \\hfill 6\\left(1\\right)+-2\\left(1\\right)+-3\\left(1\\right)\\end{array}\\right]\\hfill \\\\ & =\\left[\\begin{array}{rrr}\\hfill 1& \\hfill 0& \\hfill 0\\\\ \\hfill 0& \\hfill 1& \\hfill 0\\\\ \\hfill 0& \\hfill 0& \\hfill 1\\end{array}\\right]\\hfill \\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>

      Given the [latex]3\\times 3[\/latex] matrix [latex]A[\/latex], find the inverse.\r\n

      [latex]A = \\left[\\begin{array}{ccc} 1 & 2 & 3 \\\\ 4 & 5 & 6 \\\\ 7 & 8 & 9 \\end{array}\\right][\/latex]<\/p>\r\n[reveal-answer q=\"817649\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"817649\"]\r\n

        \r\n \t
      1. Augment the matrix [latex]A[\/latex] with the identity matrix:\r\n
        [latex]\\left[\\begin{array}{ccc|ccc} 1 & 2 & 3 & 1 & 0 & 0 \\\\ 4 & 5 & 6 & 0 & 1 & 0 \\\\ 7 & 8 & 9 & 0 & 0 & 1 \\end{array}\\right][\/latex]<\/center><\/li>\r\n \t
      2. Perform row operations to try to transform the left side into the identity matrix:\r\n
          \r\n \t
        • Subtract 4 times row 1 from row 2:\r\n
          [latex]R_2 \\leftarrow R_2 - 4R_1[\/latex]<\/center>\r\n
          [latex]\\left[\\begin{array}{ccc|ccc} 1 & 2 & 3 & 1 & 0 & 0 \\\\ 0 & -3 & -6 & -4 & 1 & 0 \\\\ 7 & 8 & 9 & 0 & 0 & 1 \\end{array}\\right][\/latex]<\/center><\/li>\r\n \t
        • Subtract 7 times row 1 from row 3:\r\n
          [latex]R_3 \\leftarrow R_3 - 7R_1[\/latex]<\/center>\r\n
          [latex]\\left[\\begin{array}{ccc|ccc} 1 & 2 & 3 & 1 & 0 & 0 \\\\ 0 & -3 & -6 & -4 & 1 & 0 \\\\ 0 & -6 & -12 & -7 & 0 & 1 \\end{array}\\right][\/latex]<\/center><\/li>\r\n \t
        • Subtract 2 times row 2 from row 3:\r\n
          [latex]R_3 \\leftarrow R_3 - 2R_2[\/latex]<\/center>\r\n
          [latex]\\left[\\begin{array}{ccc|ccc} 1 & 2 & 3 & 1 & 0 & 0 \\\\ 0 & -3 & -6 & -4 & 1 & 0 \\\\ 0 & 0 & 0 & 1 & -2 & 1 \\end{array}\\right][\/latex]<\/center><\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\nThe third row of the left side of the augmented matrix is all zeros. This indicates that the original matrix A is singular, meaning it does not have an inverse<\/strong>.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>
          How to: Finding an inverse using a TI-84 calculator<\/b>\r\n
            \r\n \t
          1. Enter the matrix in the [MATRIX] menu.<\/li>\r\n \t
          2. Find the inverse of the matrix by entering [latex][A]^-1[\/latex] into the calculator.<\/li>\r\n \t
          3. Use [MATH] + Frac to convert the inverse into fraction form.<\/li>\r\n<\/ol>\r\n<\/section>
            [ohm_question hide_question_numbers=1]321757[\/ohm_question]<\/section>
            [ohm_question hide_question_numbers=1]321758[\/ohm_question]<\/section><\/section>
            [ohm_question hide_question_numbers=1]321759[\/ohm_question]<\/section>","rendered":"

            Finding the Multiplicative Inverse of 2\u00d72 Matrices Using a Formula<\/h2>\n

            When we need to find the multiplicative inverse<\/strong> of a [latex]2\\times 2[\/latex] matrix, we can use a special formula instead of using matrix multiplication or augmenting with the identity.<\/p>\n

            \n

            multiplicative inverse of 2\u00d72 matrices formula<\/h3>\n

            If [latex]A[\/latex] is a [latex]2\\times 2[\/latex] matrix, such as<\/p>\n

            [latex]A=\\left[\\begin{array}{rrr}\\hfill a& \\hfill & \\hfill b\\\\ \\hfill c& \\hfill & \\hfill d\\end{array}\\right][\/latex]<\/p>\n

            the multiplicative inverse of [latex]A[\/latex] is given by the formula<\/p>\n

            [latex]{A}^{-1}=\\frac{1}{ad-bc}\\left[\\begin{array}{rrr}\\hfill d& \\hfill & \\hfill -b\\\\ \\hfill -c& \\hfill & \\hfill a\\end{array}\\right][\/latex]<\/p>\n

            where [latex]ad-bc\\ne 0[\/latex].<\/p>\n

             <\/p>\n

            If [latex]ad-bc=0[\/latex], then [latex]A[\/latex] has no inverse.<\/p>\n<\/section>\n

            Use the formula to find the multiplicative inverse of<\/p>\n

            [latex]A=\\left[\\begin{array}{cc}1& -2\\\\ 2& -3\\end{array}\\right][\/latex]<\/p>\n