# 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:

 $\ds \paren {\mathbf A^{-1} }^\intercal \mathbf A^\intercal$ $=$ $\ds \paren {\mathbf A \mathbf A^{-1} }^\intercal$ Transpose of Matrix Product $\ds$ $=$ $\ds \mathbf I^\intercal$ Definition of Inverse Matrix: $\mathbf I$ denotes Unit Matrix $\ds$ $=$ $\ds \mathbf I$ Definition of Unit Matrix

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

$\paren {\mathbf A^{-1} }^\intercal = \paren {\mathbf A^\intercal}^{-1}$

$\blacksquare$