Transpose of inverse vs inverse of transpose

Question

Given a square matrix, is the transpose of the inverse equal to the inverse of the transpose?

$$ (A^{-1})^T = (A^T)^{-1} $$

Answer 1

Is $(A^{-1})^T = (A^T)^{-1}$ you ask.

Well $$\begin{align} A^T(A^{-1})^T = (A^{-1}A)^{T} = I^T = I \\ (A^{-1})^TA^T = (AA^{-1})^{T} = I^T = I \end{align} $$

This proves that the inverse of $A^T$ is $(A^{-1})^T$. So the answer to your question is yes.

Here I have used that $$ A^TB^T = (BA)^T. $$ And we have used that the inverse of a matrix $A$ is exactly (by definition) the matrix $B$ such that $AB = BA = I$.

Answer 2

The result is true. See part (4) of Theorem 3.5.3 in $\textit{A Modern Introduction to Linear Algebra}$ by Henry Ricardo (CRC Press, 2010), for example.