Question

Find the matrix $A$ for which $\left[\begin{array}{cc}2& 1\\ 3& 2\end{array}\right]A\left[\begin{array}{cc}-3& 2\\ 5& -3\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$.

Answer 1

Let us take $B=\left[\begin{array}{cc}2& 1\\ 3& 2\end{array}\right]$

here $det\left(B\right)=2\left(2\right)-3\left(1\right)=1.$

So, $B^{-1}=\frac{Adj\left(B\right)}{|B|}=\frac{1}{1}\left[\begin{array}{cc}2& -1\\ -3& 2\end{array}\right]$

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

Similarly $C=\left[\begin{array}{cc}-3& 2\\ 5& -3\end{array}\right],det\left(c\right)=g-10$

$=-1$

So, $c^{-1}=\frac{Adj\left(c\right)}{|c|}=\frac{1}{-1}\left[\begin{array}{cc}-3& -2\\ -5& -3\end{array}\right]$

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

So, we get, $B.AC=I_2$

So, $B^{-1}(B.A.C).C^{-1}=B^{-1}.I_2.C^{-1}$

$(B^{-1}.B).\left(A\right).(C.C^{-1})=B^{-1}.C^{-1}$

$I_2.A.I_2=B^{-1}.C^{-1}$

$A+B^{-1}.C^{-1}$

So,

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

$=\left[\begin{array}{cc}2\left(3\right)-1\left(5\right)& 2\left(2\right)-1\left(3\right)\\ -3\left(3\right)+2\left(5\right)& -3\left(2\right)+2\left(3\right)\end{array}\right]$

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