Question

Given A = $\left[\begin{array}{cc}2& -3\\ -4& 7\end{array}\right]$, compute $A^{-1}$ and show that $2A^{-1}=9I-A$.

Answer 1

Here A = $\left[\begin{array}{cc}2& -3\\ -4& 7\end{array}\right]$ $\Rightarrow$ adj.A = $\left[\begin{array}{cc}7& 3\\ 4& 2\end{array}\right]$ and |A| = $\left[\begin{array}{cc}2& -3\\ -4& 7\end{array}\right]$ = 14-12= 2

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

Now $2A^{-1}=\left[\begin{array}{cc}7& 3\\ 4& 2\end{array}\right]$ and , $9I-A=9$ $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]-\left[\begin{array}{cc}2& -3\\ -4& 7\end{array}\right]=\left[\begin{array}{cc}9& 0\\ 0& 9\end{array}\right]-\left[\begin{array}{cc}2& -3\\ -4& 7\end{array}\right]=\left[\begin{array}{cc}7& 3\\ 4& 2\end{array}\right]$

Clearly , $2A^{-1}=9I-A$.