Chapter 7.3, Problem 9QR

Solution

To identify: The inverse of the matrix.

  $\left[\begin{array}{cc}1& 3\\ -2& -2\end{array}\right]$

  $A^{-1}=\left[\begin{array}{cc}\frac{-1}{2}& -\frac{3}{4}\\ \frac{1}{2}& \frac{1}{4}\end{array}\right]$

Given information:

The given matrix is.

  $\left[\begin{array}{cc}1& 3\\ -2& -2\end{array}\right]$

Explanation:

Consider the given information.

In order to determine the inverse of the given matrix, first compute its determinant.

  $Det\left(A\right)=1\left(-2\right)-3\left(-2\right)=4$

  $A^{-1}=\frac{1}{DetA}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$

Since the determinants is not zero the matrix has an inverse in the form.

  $\begin{array}{l}{A}^{-1}=\frac{1}{4}\left[\begin{array}{cc}-2& -3\\ 2& 1\end{array}\right]\\ A^{-1}=\left[\begin{array}{cc}\frac{-1}{2}& -\frac{3}{4}\\ \frac{1}{2}& \frac{1}{4}\end{array}\right]\end{array}$

Hence, the inverse of the matrix $\left[\begin{array}{cc}1& 3\\ -2& -2\end{array}\right]$ is $\left[\begin{array}{cc}\frac{-1}{2}& -\frac{3}{4}\\ \frac{1}{2}& \frac{1}{4}\end{array}\right]$ .