Chapter 3.8, Problem 7CYU

To determine

To find: the inverse of matrix $\left[\begin{array}{cc}-3& 0\\ 5& 2\end{array}\right]$ .

Answer to Problem 7CYU

The inverse of matrix is $\frac{1}{6}\left[\begin{array}{cc}-2& 0\\ 5& 3\end{array}\right]$ .

Explanation of Solution

Given:

The matrix is $\left[\begin{array}{cc}-3& 0\\ 5& 2\end{array}\right]$ .

Calculation:

Write the expression of inverse of matrix as follows:

Consider the matrix A ,

  $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

Inverse of matrix is,

  $\begin{array}{c}{A}^{-1}=\frac{1}{\left|A\right|}\left(adjA\right)\\ =\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]\end{array}$

Consider matrix and determine the inverse as follows:

  $\begin{array}{c}{A}^{-1}=\frac{1}{\left|A\right|}\left(adjA\right)\\ =\frac{1}{\left(-3\right)\times \left(2\right)-\left(0\right)\times \left(5\right)}\left[\begin{array}{cc}2& 0\\ -5& -3\end{array}\right]\\ =\frac{-1}{6}\left[\begin{array}{cc}2& 0\\ -5& -3\end{array}\right]\\ =\frac{1}{6}\left[\begin{array}{cc}-2& 0\\ 5& 3\end{array}\right]\end{array}$

Thus, the inverse of matrix is $\frac{1}{6}\left[\begin{array}{cc}-2& 0\\ 5& 3\end{array}\right]$ .