Chapter 3, Problem 17PT

To determine

To find the inverse of the matrix $\left[\begin{array}{cc}5& 0\\ 0& 1\end{array}\right]$ , if it exists.

Answer to Problem 17PT

  $A^{-1}=\left[\begin{array}{cc}\frac{1}{5}& 0\\ 0& 1\end{array}\right]$

Explanation of Solution

Given information :

Given matrix $\left[\begin{array}{cc}5& 0\\ 0& 1\end{array}\right]$

Let $A=\left[\begin{array}{cc}5& 0\\ 0& 1\end{array}\right]$

  $\begin{array}{l}\text{Since,}\left|A\right|=\left|\begin{array}{cc}5& 0\\ 0& 1\end{array}\right|\\ =5-0\\ \therefore A=5\ne 0\\ \Rightarrow A^{-1}\text{Exists}\text{.}\end{array}$

  $\begin{array}{l}Now{A}^{-1}=\frac{adjA}{\left|A\right|},\text{where}adjA=\left[\begin{array}{cc}1& 0\\ 0& 5\end{array}\right]\\ \Rightarrow A^{-1}=\frac{1}{5}\left[\begin{array}{cc}1& 0\\ 0& 5\end{array}\right]\\ \therefore A^{-1}=\left[\begin{array}{cc}\frac{1}{5}& 0\\ 0& 1\end{array}\right]\end{array}$