Inverse of Transpose of Matrix is Transpose of Inverse

Theorem
Let $\mathbf A$ be a matrix over a field.

Let $\mathbf A^\intercal$ denote the transpose of $\mathbf A$.

Let $\mathbf A$ be an invertible matrix.

Then $\mathbf A^\intercal$ is also invertible and:
 * $\paren {\mathbf A^\intercal}^{-1} = \paren {\mathbf A^{-1} }^\intercal$

where $\mathbf A^{-1}$ denotes the inverse of $\mathbf A$.

Proof
We have:

Hence $\paren {\mathbf A^{-1} }^\intercal$ is an inverse of $\mathbf A^\intercal$.

From Inverse of Square Matrix over Field is Unique:
 * $\paren {\mathbf A^{-1} }^\intercal = \paren {\mathbf A^\intercal}^{-1}$