Question

Find the inverse of the matrix $\left[\begin{array}{cc}-3& 2\\ 5& -3\end{array}\right].$ Hence, find the matrix P satisfying the matrix equation $P\left[\begin{array}{cc}-3& 2\\ 5& -3\end{array}\right]=\left[\begin{array}{cc}1& 2\\ 2& -1\end{array}\right].$

Answer 1

Let $A=\left[\begin{array}{cc}-3& 2\\ 5& -3\end{array}\right]\Rightarrow |A|=\left[\begin{array}{cc}-3& 2\\ 5& -3\end{array}\right]=9-10=-1,adj.A=\left[\begin{array}{cc}-3& -2\\ -5& -3\end{array}\right]$
$\therefore A^{-1}=\frac{adj.A}{|A|}=\left[\begin{array}{cc}3& 2\\ 5& 3\end{array}\right]...\left(i\right)$
Now $P\left[\begin{array}{cc}-3& 2\\ 5& -3\end{array}\right]=\left[\begin{array}{cc}1& 2\\ 2& -1\end{array}\right]\Rightarrow P=\left[\begin{array}{cc}1& 2\\ 2& -1\end{array}\right]{\left[\begin{array}{cc}-3& 2\\ 5& -3\end{array}\right]}^{-1}$ (On post-multiplication with $A^{-1}.$
By (i), we get : $P=\left[\begin{array}{cc}1& 2\\ 2& -1\end{array}\right]\left[\begin{array}{cc}3& 2\\ 5& 3\end{array}\right]$ $\therefore P=\left[\begin{array}{cc}3+10& 2+6\\ 6-5& 4-3\end{array}\right]=\left[\begin{array}{cc}13& 8\\ 1& 1\end{array}\right]$