{"id":1378,"date":"2025-07-24T19:14:21","date_gmt":"2025-07-24T19:14:21","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1378"},"modified":"2025-08-13T02:34:13","modified_gmt":"2025-08-13T02:34:13","slug":"solving-systems-with-inverses-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/solving-systems-with-inverses-learn-it-2\/","title":{"raw":"Solving Systems with Inverses: Learn It 2","rendered":"Solving Systems with Inverses: Learn It 2"},"content":{"raw":"

Finding the Multiplicative Inverse<\/h2>\r\nWe can now determine whether two matrices are inverses, but how would we find the inverse of a given matrix?\r\n\r\nThere are several methods to accomplish this, each with its own advantages and applications. Understanding these techniques will provide you with a solid foundation for solving systems of linear equations and other matrix-related problems. Let's dive into the different approaches to finding the inverse of a matrix!\r\n

Matrix Multiplication<\/h3>\r\nSince we know that the product of a matrix and its inverse is the identity matrix, we can find the inverse of a matrix by setting up an equation using\u00a0matrix multiplication<\/strong>.\r\n\r\n
Use matrix multiplication to find the inverse of the given matrix.\r\n

[latex]A=\\left[\\begin{array}{rrr}\\hfill 1& \\hfill & \\hfill -2\\\\ \\hfill 2& \\hfill & \\hfill -3\\end{array}\\right][\/latex]<\/p>\r\n[reveal-answer q=\"31546\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"31546\"]\r\n\r\nFor this method, we multiply [latex]A[\/latex] by a matrix containing unknown constants and set it equal to the identity.\r\n

[latex]\\left[\\begin{array}{rr}\\hfill 1& \\hfill -2\\\\ \\hfill 2& \\hfill -3\\end{array}\\right]\\text{ }\\left[\\begin{array}{rr}\\hfill a& \\hfill b\\\\ \\hfill c& \\hfill d\\end{array}\\right]=\\left[\\begin{array}{rr}\\hfill 1& \\hfill 0\\\\ \\hfill 0& \\hfill 1\\end{array}\\right][\/latex]<\/p>\r\nFind the product of the two matrices on the left side of the equal sign.\r\n

[latex]\\left[\\begin{array}{rr}\\hfill 1& \\hfill -2\\\\ \\hfill 2& \\hfill -3\\end{array}\\right]\\text{ }\\left[\\begin{array}{rr}\\hfill a& \\hfill b\\\\ \\hfill c& \\hfill d\\end{array}\\right]=\\left[\\begin{array}{rr}\\hfill 1a - 2c& \\hfill 1b - 2d\\\\ \\hfill 2a - 3c& \\hfill 2b - 3d\\end{array}\\right][\/latex]<\/p>\r\nNext, set up a system of equations with the entry in row 1, column 1 of the new matrix equal to the first entry of the identity, 1. Set the entry in row 2, column 1 of the new matrix equal to the corresponding entry of the identity, which is 0.\r\n

[latex]\\begin{array}{c}1a - 2c=1\\text{ }{R}_{1}\\\\ 2a - 3c=0\\text{ }{R}_{2}\\end{array}[\/latex]<\/p>\r\nUsing row operations, multiply and add as follows: [latex]\\left(-2\\right){R}_{1}+{R}_{2}\\to {R}_{2}[\/latex]. Add the equations, and solve for [latex]c[\/latex].\r\n

[latex]\\begin{array}{r}\\hfill 1a - 2c=1\\\\ \\hfill 0+1c=-2\\\\ \\hfill c=-2\\end{array}[\/latex]<\/p>\r\nBack-substitute to solve for [latex]a[\/latex].\r\n

[latex]\\begin{array}{r}\\hfill a - 2\\left(-2\\right)=1\\\\ \\hfill a+4=1\\\\ \\hfill a=-3\\end{array}[\/latex]<\/p>\r\nWrite another system of equations setting the entry in row 1, column 2 of the new matrix equal to the corresponding entry of the identity, 0. Set the entry in row 2, column 2 equal to the corresponding entry of the identity.\r\n

[latex]\\begin{array}{rr}\\hfill 1b - 2d=0& \\hfill {R}_{1}\\\\ \\hfill 2b - 3d=1& \\hfill {R}_{2}\\end{array}[\/latex]<\/p>\r\nUsing row operations, multiply and add as follows: [latex]\\left(-2\\right){R}_{1}+{R}_{2}={R}_{2}[\/latex]. Add the two equations and solve for [latex]d[\/latex].\r\n

[latex]\\begin{array}{r}\\hfill 1b - 2d=0\\\\ \\hfill \\frac{0+1d=1}{d=1}\\\\ \\hfill \\end{array}[\/latex]<\/p>\r\nOnce more, back-substitute and solve for [latex]b[\/latex].\r\n

[latex]\\begin{array}{r}\\hfill b - 2\\left(1\\right)=0\\\\ \\hfill b - 2=0\\\\ \\hfill b=2\\end{array}[\/latex]<\/p>\r\n

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

Augmenting with the Identity<\/h3>\r\nAnother way to find the multiplicative inverse<\/strong> is by augmenting with the identity. When matrix [latex]A[\/latex] is transformed into [latex]I[\/latex], the augmented matrix [latex]I[\/latex] transforms into [latex]{A}^{-1}[\/latex].\r\n\r\n
For example, given\r\n

[latex]A=\\left[\\begin{array}{rrr}\\hfill 2& \\hfill & \\hfill 1\\\\ \\hfill 5& \\hfill & \\hfill 3\\end{array}\\right][\/latex]<\/p>\r\naugment [latex]A[\/latex] with the identity\r\n

[latex]\\left[\\begin{array}{cc|cc}\\hfill 2& \\hfill 1& \\hfill 1& \\hfill 0\\\\ \\hfill 5& \\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. Switch row 1 and row 2.\r\n
    [latex]\\left[\\begin{array}{cc|cc}\\hfill 5& \\hfill 3& \\hfill 0& \\hfill 1\\\\ \\hfill 2& \\hfill 1& \\hfill 1& \\hfill 0\\\\ \\end{array}\\right][\/latex]<\/center><\/li>\r\n \t
  2. Multiply row 2 by [latex]-2[\/latex] and add to row 1.\r\n
    [latex]\\left[\\begin{array}{cc|cc}\\hfill 1& \\hfill 1& \\hfill -2& \\hfill 1\\\\ \\hfill 2& \\hfill 1& \\hfill 1& \\hfill 0\\\\ \\end{array}\\right][\/latex]<\/center><\/li>\r\n \t
  3. Multiply row 1 by [latex]-2[\/latex] and add to row 2.\r\n
    [latex]\\left[\\begin{array}{cc|cc}\\hfill 1& \\hfill 1& \\hfill -2& \\hfill 1\\\\ \\hfill 0& \\hfill -1& \\hfill 5& \\hfill -2\\\\ \\end{array}\\right][\/latex]<\/center><\/li>\r\n \t
  4. Add row 2 to row 1.\r\n
    [latex]\\left[\\begin{array}{cc|cc}\\hfill 1& \\hfill 0& \\hfill 3& \\hfill -1\\\\ \\hfill 0& \\hfill -1& \\hfill 5& \\hfill -2\\\\ \\end{array}\\right][\/latex]<\/center><\/li>\r\n \t
  5. Multiply row 2 by [latex]-1[\/latex].\r\n
    [latex]\\left[\\begin{array}{cc|cc}\\hfill 1& \\hfill 0& \\hfill 3& \\hfill -1\\\\ \\hfill 0& \\hfill 1& \\hfill -5& \\hfill 2\\\\ \\end{array}\\right][\/latex]<\/center><\/li>\r\n<\/ol>\r\nThe matrix we have found is [latex]{A}^{-1}[\/latex].\r\n

    [latex]{A}^{-1}=\\left[\\begin{array}{rrr}\\hfill 3& \\hfill & \\hfill -1\\\\ \\hfill -5& \\hfill & \\hfill 2\\end{array}\\right][\/latex]<\/p>\r\n\r\n<\/section>

    Find the inverse of the given matrix by augmenting with the identity.\r\n

    [latex]A=\\left[\\begin{array}{rrr}\\hfill 1& \\hfill & \\hfill -2\\\\ \\hfill 2& \\hfill & \\hfill -3\\end{array}\\right][\/latex]<\/p>\r\n[reveal-answer q=\"967719\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"967719\"]\r\n\r\n[latex]\\begin{array}{l} \\text{1. Augment } A \\text{ with the identity matrix:} \\\\ \\left[\\begin{array}{cc|cc} 1 & -2 & 1 & 0 \\\\ 2 & -3 & 0 & 1 \\\\ \\end{array}\\right] \\\\ \\\\ \\text{2. Perform row operations with the goal of turning } A \\text{ into the identity matrix.} \\\\ \\text{First, multiply row 1 by } -2 \\text{ and add to row 2:} \\\\ \\left[\\begin{array}{cc|cc} 1 & -2 & 1 & 0 \\\\ 2 & -3 & 0 & 1 \\\\ \\end{array}\\right] \\rightarrow \\left[\\begin{array}{cc|cc} 1 & -2 & 1 & 0 \\\\ 0 & 1 & -2 & 1 \\\\ \\end{array}\\right] \\\\ \\\\ \\text{Next, multiply row 2 by } 2 \\text{ and add to row 1:} \\\\ \\left[\\begin{array}{cc|cc} 1 & -2 & 1 & 0 \\\\ 0 & 1 & -2 & 1 \\\\ \\end{array}\\right] \\rightarrow \\left[\\begin{array}{cc|cc} 1 & 0 & -3 & 2 \\\\ 0 & 1 & -2 & 1 \\\\ \\end{array}\\right] \\\\ \\\\ \\text{3. The matrix we have found is } A^{-1} : \\\\ A^{-1} = \\left[\\begin{array}{rr} -3 & 2 \\\\ -2 & 1 \\\\ \\end{array}\\right] \\\\ \\end{array}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n

    <\/h3>","rendered":"

    Finding the Multiplicative Inverse<\/h2>\n

    We can now determine whether two matrices are inverses, but how would we find the inverse of a given matrix?<\/p>\n

    There are several methods to accomplish this, each with its own advantages and applications. Understanding these techniques will provide you with a solid foundation for solving systems of linear equations and other matrix-related problems. Let’s dive into the different approaches to finding the inverse of a matrix!<\/p>\n

    Matrix Multiplication<\/h3>\n

    Since we know that the product of a matrix and its inverse is the identity matrix, we can find the inverse of a matrix by setting up an equation using\u00a0matrix multiplication<\/strong>.<\/p>\n

    Use matrix multiplication to find the inverse of the given matrix.<\/p>\n

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