Question
If $A=\left[\begin{array}{cc}2 & 2 \\ -3 & 2\end{array}\right]$ and $B=\left[\begin{array}{cc}0 & -1 \\ 1 & 0\end{array}\right]$, then find the matrix $\left(B^{-1}\right.$ $\left.A^{-1}\right)^{-1}$
✓
Answer
$\left(B^{-1} A^{-1}\right)^{-1}=\left[(A B)^{-1}\right]^{-1} \quad \ldots \ldots . .\left[\left(A B^{-1}\right)=B^{-1} A^{-1}\right] $
$ =A B$
$ \therefore\left(B^{-1} A^{-1}\right)^{-1}=\left[\begin{array}{cc}2 & 2 \\ -3 & 2\end{array}\right]\left[\begin{array}{cc}0 & -1 \\ 1 & 0\end{array}\right] $
$ =\left[\begin{array}{ll}0+2 & -2+0 \\ 0+2 & 3+0\end{array}\right] $
$ =\left[\begin{array}{cc}2 & -2 \\ 2 & 3\end{array}\right]$
$ =A B$
$ \therefore\left(B^{-1} A^{-1}\right)^{-1}=\left[\begin{array}{cc}2 & 2 \\ -3 & 2\end{array}\right]\left[\begin{array}{cc}0 & -1 \\ 1 & 0\end{array}\right] $
$ =\left[\begin{array}{ll}0+2 & -2+0 \\ 0+2 & 3+0\end{array}\right] $
$ =\left[\begin{array}{cc}2 & -2 \\ 2 & 3\end{array}\right]$
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