Question

Find the matrix $A$ satisfying the matrix equation:

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

Answer 1

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

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

Then given matrix equation become $PAQ=I$

Pre multiply $P^{-1}$ to both sides, we get

$P^{-1}PAQ=P^{-1}\cdot I$

$\Rightarrow IAQ=P^{-1}$

$\Rightarrow AQ=P^{-1}$

$\Rightarrow AQ{Q}^{-1}=P^{-1}{Q}^{-1}$

[Post multiply by $Q^{-1}$ to both sides]

$\Rightarrow AI=P^{-1}{Q}^{-1}$

$\Rightarrow A=P^{-1}{Q}^{-1}$ ...$\left(i\right)$

Finding $P^{-1}$ and $Q^{-1}$

$P=\left[\begin{array}{cc}2& 1\\ 3& 2\end{array}\right]$

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

And $|P|=1$

$\therefore P^{-1}=\frac{adjP}{|P|}=\left[\begin{array}{cc}2& -1\\ -3& 2\end{array}\right]$ ...$\left(ii\right)$

And $Q=\left[\begin{array}{cc}-3& 2\\ 5& -3\end{array}\right]$

Now $adjQ=\left[\begin{array}{cc}-3& -2\\ -5& -3\end{array}\right]$

And $|Q|=-1$

$\therefore Q^{-1}=\frac{adjQ}{|Q|}=\left[\begin{array}{cc}3& 2\\ 5& 3\end{array}\right]$ ....$\left(iii\right)$

Finding $A$ using $\left(i\right)$

As $A=P^{-1}{Q}^{-1}$

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

[from equation $\left(ii\right)$ and $\left(iii\right)$]

$\Rightarrow A=\left[\begin{array}{cc}6-5& 4-3\\ -9+10& -6+6\end{array}\right]$

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

Hence, matrix $A=\left[\begin{array}{cc}1& 1\\ 1& 0\end{array}\right]$