Question

If A is a non-singular square matrix such that \[A^{- 1} = \begin{bmatrix}5 & 3 \\ - 2 & - 1\end{bmatrix}\] , then find \[\left( A^T \right)^{- 1} .\]

Answer 1

For any invertible matrix A, \[( A^T )^{- 1} = ( A^{- 1} )^T\] 
We have
\[ A^{- 1} = \begin{bmatrix}5 & 3 \\ - 2 & - 1\end{bmatrix}\]
\[ \Rightarrow ( A^T )^{- 1} = \begin{bmatrix}5 & - 2 \\ 3 & - 1\end{bmatrix}\]