Chapter 10.5, Problem 17E

To determine

To calculate: The inverse ofmatrix if exist.

Answer to Problem 17E

Inverse of matrix is $A=\left[\begin{array}{ccc}2& 4& 1\\ -1& 1& -1\\ 1& 4& 0\end{array}\right]$ is $A^{-1}=\left[\begin{array}{ccc}-4& -4& 5\\ 1& 1& -1\\ 5& 4& -6\end{array}\right]$

Explanation of Solution

Given information:

  $\left[\begin{array}{ccc}2& 4& 1\\ -1& 1& -1\\ 1& 4& 0\end{array}\right]$

Calculation:

Let call the given matrix as $A$ as,

  $A=\left[\begin{array}{ccc}2& 4& 1\\ -1& 1& -1\\ 1& 4& 0\end{array}\right]$

Dimension of the matrix $A$ is $3\times 3$

To find the Inverse of matrix, write the augmented matrix with left side as matrix $A$ and right side as identity matrix.

Then perform the row operations on this matrix to get augmented matrix with left side as identity matrix and right side as inverse of $A$ .

  $\left[A/I\right]=\left[\begin{array}{ccc}2& 4& 1\\ -1& 1& -1\\ 1& 4& 0\end{array}|\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

Apply $R_1\leftrightarrow R_3$

  $\left[A/I\right]=\left[\begin{array}{ccc}1& 4& 0\\ -1& 1& -1\\ 2& 4& 1\end{array}|\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right]$

Apply $R_1+R_2\to R_2,R_3-2R_1\to R_3$

  $\left[A/I\right]=\left[\begin{array}{ccc}1& 4& 0\\ 0& 5& -1\\ 0& -4& 1\end{array}|\begin{array}{ccc}0& 0& 1\\ 0& 1& 1\\ 1& 0& -2\end{array}\right]$

Apply $R_2+R_3\to R_3$ ,

  $\left[A/I\right]=\left[\begin{array}{ccc}1& 4& 0\\ 0& 1& 0\\ 0& -4& 1\end{array}|\begin{array}{ccc}0& 0& 1\\ 1& 1& -1\\ 1& 0& -2\end{array}\right]$

Apply $R_3+4R_2\to R_3$

  $\left[A/I\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}|\begin{array}{ccc}0& 0& 1\\ 1& 1& -1\\ 5& 4& -6\end{array}\right]$

Apply $R_1-4R_2\to R_1$

  $\begin{array}{l}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}|\begin{array}{ccc}-4& -4& 5\\ 1& 1& -1\\ 5& 4& -6\end{array}\right]\\ =\left[I|A^{-1}\right]\end{array}$

Inverse of matrix is $A$ is $A^{-1}=\left[\begin{array}{ccc}-4& -4& 5\\ 1& 1& -1\\ 5& 4& -6\end{array}\right]$