Matrix transpose and inversion

Jan 9, 2017 • linear algebra • 0

Prove for a nonsingular (e.g. square and invertible) matrix that $(A^T)^{-1} = (A^{-1})^T$ .

We have \begin{align} I = (AA^{-1})^T = (A^{-1})^T A^T. \end{align}

Multiplying both sides by $(A^T)^{-1}$ gives

\begin{align} (A^T)^{-1} = (A^{-1})^T A^T (A^T)^{-1} = (A^{-1})^T I = (A^{-1})^T. \end{align}