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 ${\left(A^T\right)}^{-1}.$

Answer 1

For any invertible matrix A,
$(A^T{)}^{-1}=(A^{-1}{)}^T$

$$\begin{gathered} \mathrm{We}\mathrm{have} \\ {A}^{-1}=\left[\begin{array}{cc}5& 3\\ -2& -1\end{array}\right] \\ \Rightarrow (A^T{)}^{-1}=\left[\begin{array}{cc}5& -2\\ 3& -1\end{array}\right] \end{gathered}$$