Question

If $\left[\begin{array}{cc}2& 1\\ 3& 2\end{array}\right]A\left[\begin{array}{cc}-3& 2\\ 5& -3\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],$ then the sum of all the elements of matrix $A$ is

Answer 1

$\left[\begin{array}{cc}2& 1\\ 3& 2\end{array}\right]A\left[\begin{array}{cc}-3& 2\\ 5& -3\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

$\Rightarrow A={\left[\begin{array}{cc}2& 1\\ 3& 2\end{array}\right]}^{-1}\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]{\left[\begin{array}{cc}-3& 2\\ 5& -3\end{array}\right]}^{-1}$

$\Rightarrow A=\left[\begin{array}{cc}2& -1\\ -3& 2\end{array}\right]\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{cc}3& 2\\ 5& 3\end{array}\right]$

$\Rightarrow A=\left[\begin{array}{cc}2& -1\\ -3& 2\end{array}\right]\left[\begin{array}{cc}3& 2\\ 5& 3\end{array}\right]=\left[\begin{array}{cc}1& 1\\ 1& 0\end{array}\right]$
$\therefore \text{Sum}\left(A\right)=3$