Question

Find the inverse of the matrix A=$\left[\begin{array}{cc}2& -4\\ 3& -5\end{array}\right]$

Answer 1

$\left|A\right|=\left[\begin{array}{cc}2& -4\\ 3& -5\end{array}\right]$ = -10+12=2$\ne 0$

$\therefore A^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]=\frac{1}{2}\left[\begin{array}{cc}-5& 4\\ -3& 2\end{array}\right]$

=$\left[\begin{array}{cc}\frac{-5}{2}& 2\\ \frac{-3}{2}& 1\end{array}\right]$