Chapter 3.8, Problem 14PPS

To determine

To find: Whether the pair of matrices $M=\left[\begin{array}{cc}0& 2\\ 4& 5\end{array}\right]$ and $N=\left[\begin{array}{cc}1& 1\\ 0& 0\end{array}\right]$ are inverse to each other or not.

Answer to Problem 14PPS

The pair of matrices is not inverse to each other.

Explanation of Solution

Given:

Two matrices are $M=\left[\begin{array}{cc}0& 2\\ 4& 5\end{array}\right]$ and $N=\left[\begin{array}{cc}1& 1\\ 0& 0\end{array}\right]$ .

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 first matrix and determine the inverse as follows:

  $\begin{array}{c}{M}^{-1}=\frac{1}{\left|M\right|}\left(adjM\right)\left[\begin{array}{cc}0& 2\\ 4& 5\end{array}\right]\\ =\frac{1}{\left(0\right)\times \left(5\right)-\left(2\right)\times \left(4\right)}\left[\begin{array}{cc}5& -2\\ -4& 0\end{array}\right]\\ =\frac{-1}{8}\left[\begin{array}{cc}5& -2\\ -4& 0\end{array}\right]\\ =\frac{1}{8}\left[\begin{array}{cc}-5& 2\\ 4& 0\end{array}\right]\end{array}$

Thus, the inverse of matrix M is $\frac{1}{8}\left[\begin{array}{cc}-5& 2\\ 4& 0\end{array}\right]$ which not equal to given matrix N . Thus, the pair of matrices is not inverse to each other.