$\text{ If }A=\left[\begin{array}{l}3\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}2\\ 4\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}3\end{array}\right]\text{ and }B=\left[\begin{array}{cc}-1& 7\\ 3& 5\end{array}\right]\text{ , then find the sum of the absolute value of the entries of }$$\text{ the matrices }X\text{ and }Y\text{ satisfying }AX=B\text{ and }YA=B\text{ is }$

$\begin{array}{l}\text{ Given that }A=\left[\begin{array}{ll}3& 2\\ 4& 3\end{array}\right]\text{ and }B=\left[\begin{array}{cc}-1& 7\\ 3& 5\end{array}\right]\\ \left|A\right|=\left|\begin{array}{ll}3& 2\\ 4& 3\end{array}\right|=9-8=1\\ A^{-1}=\frac{\mathrm{adj}\left(A\right)}{\left|A\right|}=\left[\begin{array}{cc}3& -2\\ -4& 3\end{array}\right]=\left[\begin{array}{cc}3& -2\\ -4& 3\end{array}\right]\\ AX=B\end{array}$

$\begin{array}{l}\Rightarrow X=A^{-1}B=\left[\begin{array}{cc}3& -2\\ -4& 3\end{array}\right]\left[\begin{array}{cc}-1& 7\\ 3& 5\end{array}\right]=\left[\begin{array}{cc}-9& 11\\ 13& -13\end{array}\right]\\ YA=B\\ \Rightarrow YA{A}^{-1}=B{A}^{-1}\\ \Rightarrow Y=B{A}^{-1}\\ =\left[\begin{array}{cl}-1& 7\\ 3& 5\end{array}\right]\left[\begin{array}{cc}3& -2\\ -4& 3\end{array}\right]\\ =\left[\begin{array}{cc}-31& 23\\ -11& 9\end{array}\right]\end{array}$

sum of the absolute values of entries of the matrices X and Y = 9+11+13+13+31+23+11+9=120