Question

Find the inverse of the given matrix
$\left[\begin{array}{cc}-1& 5\\ -3& 2\end{array}\right]$

Answer 1

Let $A=\left[\begin{array}{cc}-1& 5\\ -3& 2\end{array}\right]$, We have, $|A|=\left|\begin{array}{cc}-1& 5\\ -3& 2\end{array}\right|=-2-(-15)=13$
Now, cofactors of A are $A_{11}=2,A_{12}=3,A_{21}=-5,A_{22}=-1$
$\therefore adj\left(A\right)={\left[\begin{array}{cc}2& 3\\ -5& -1\end{array}\right]}^T=\left[\begin{array}{cc}2& -5\\ 3& -1\end{array}\right]$
Now, $A^{-1}=\frac{1}{|A|}\left(adjA\right)=\frac{1}{13}\left[\begin{array}{cc}2& -5\\ 3& -1\end{array}\right]=\left[\begin{array}{cc}\frac{2}{13}& -\frac{5}{13}\\ \frac{3}{13}& -\frac{1}{13}\end{array}\right]$