Find the matrix A satisfying
Question:

Find the matrix A satisfying the matrix equation:

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

Solution:

Given,

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

Let $\mathrm{P}=\left[\begin{array}{ll}2 & 1 \\ 3 & 2\end{array}\right]_{\mathrm{Q}}=\left[\begin{array}{cc}-3 & 2 \\ 5 & -3\end{array}\right]$ and $\mathrm{I}=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$

Now,

P-1 PAQ = P-1 I

So, IAQ = P-1

AQ = P-1

AQQ-1 = P-1 Q-1

AI = P-1 Q-1

A = P-1 Q-1

Now adj. $P=\left[\begin{array}{cc}2 & -1 \\ -3 & 2\end{array}\right]$ and $|P|=1$

Hence, $\quad P^{-1}=\left[\begin{array}{cc}2 & -1 \\ -3 & 2\end{array}\right]$

Also adj. $Q=\left[\begin{array}{rr}-3 & -2 \\ -5 & -3\end{array}\right]$ and $|Q|=-1$

Hence, $\quad Q^{-1}=\left[\begin{array}{ll}3 & 2 \\ 5 & 3\end{array}\right]$

Thus, $A=P^{-1} Q^{-1}$

$=\left[\begin{array}{cc}2 & -1 \\ -3 & 2\end{array}\right]\left[\begin{array}{ll}3 & 2 \\ 5 & 3\end{array}\right]=\left[\begin{array}{cc}6-5 & 4-3 \\ -9+10 & -6+6\end{array}\right]=\left[\begin{array}{ll}1 & 1 \\ 1 & 0\end{array}\right]$

$\left[\begin{array}{l}4 \\ 1 \\ 3\end{array}\right] A=\left[\begin{array}{lll}-4 & 8 & 4 \\ -1 & 2 & 1 \\ -3 & 6 & 3\end{array}\right]$