Chapter 3, Problem 43SGR

To determine

To find the inverse of the matrix $A=\left[\begin{array}{cc}7& 4\\ 3& 2\end{array}\right]$ , if it exists.

Answer to Problem 43SGR

  $A^{-1}=\left[\begin{array}{cc}1& -2\\ -\frac{3}{2}& \frac{7}{2}\end{array}\right]$

Explanation of Solution

Given information :

Given matrix $A=\left[\begin{array}{cc}7& 4\\ 3& 2\end{array}\right]$

  $\begin{array}{l}\text{Since,}\left|A\right|=\left|\begin{array}{cc}7& 4\\ 3& 2\end{array}\right|\\ =\left(\left(7\right)\times 2\right)-\left(\left(4\right)\times \left(3\right)\right)\\ =14-12\\ \therefore \left|A\right|=2\ne 0.\\ \Rightarrow A^{-1}\text{Exists}\text{.}\end{array}$

  $\begin{array}{l}Now{A}^{-1}=\frac{adjA}{\left|A\right|},\text{where}adjA=\left[\begin{array}{cc}2& -4\\ -3& 7\end{array}\right]\\ \Rightarrow A^{-1}=\frac{1}{2}\left[\begin{array}{cc}2& -4\\ -3& 7\end{array}\right]\\ \therefore A^{-1}=\left[\begin{array}{cc}1& -2\\ -\frac{3}{2}& \frac{7}{2}\end{array}\right]\end{array}$