Question

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

Answer 1

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

$|A|=14-12=2$

$A^{-1}=\frac{cofactormatrixofA}{|A|}$

$=\frac{1}{2}\left[\begin{array}{cc}7& 3\\ 4& 2\end{array}\right]$

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

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

$=2A^{-1}$

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