Question

Given the matrices $A=\left[\begin{array}{cc}3& 2\\ 4& 3\end{array}\right]$ and $B=\left[\begin{array}{cc}-1& 7\\ 3& 5\end{array}\right].$ Then sum of the absolute values of the entries of the matrices $X$ and $Y$ satisfying $AX=B$ and $YA=B$ is

Answer 1

Matrix $A$ is non-singular, and $det\left(A\right)=1$
$A^{-1}=\frac{adjA}{|A|}=\frac{\left[\begin{array}{cc}3& -2\\ -4& 3\end{array}\right]}{1}$
$\Rightarrow A^{-1}=\left[\begin{array}{cc}3& -2\\ -4& 3\end{array}\right]$
We have, $AX=B$
$\Rightarrow X=A^{-1}B$

And $YA=B$
$\Rightarrow Y=B{A}^{-1}$

Hence, $X=\left[\begin{array}{cc}3& -2\\ -4& 3\end{array}\right]\left[\begin{array}{cc}-1& 7\\ 3& 5\end{array}\right]=\left[\begin{array}{cc}-9& 11\\ 13& -13\end{array}\right]$
$Y=\left[\begin{array}{cc}-1& 7\\ 3& 5\end{array}\right]\left[\begin{array}{cc}3& -2\\ -4& 3\end{array}\right]=\left[\begin{array}{cc}-31& 23\\ -11& 9\end{array}\right]$
Sum of the absolute values of the entries $=9+11+13+13+31+23+11+9=120$