Question:
If $A$ is a non-singular square matrix such that $A^{-1}=\left[\begin{array}{cc}5 & 3 \\ -2 & -1\end{array}\right]$, then find $\left(A^{T}\right)^{-1} .$
Solution:
For any invertible matrix A,
$\left(A^{T}\right)^{-1}=\left(A^{-1}\right)^{T}$
We have
$A^{-1}=\left[\begin{array}{cc}5 & 3 \\ -2 & -1\end{array}\right]$
$\Rightarrow\left(A^{T}\right)^{-1}=\left[\begin{array}{ll}5 & -2 \\ 3 & -1\end{array}\right]$