ThmDex – An index of mathematical definitions, results, and conjectures.

Result R5500 on D5871: Invertible real matrix

Transpose of real matrix inverse is inverse to the transpose

Formulation 0

Let $A \in \mathbb{R}^{N \times N}$ be an D4571: Real matrix such that

(i)	$A$ is an D5871: Invertible real matrix
(ii)	$B$ is an D2089: Inverse matrix for $A$

Then

(1)	$A^T$ is an D5871: Invertible real matrix
(2)	$B^T$ is an D2089: Inverse matrix for $A^T$

Proofs

Proof 0

Let $A \in \mathbb{R}^{N \times N}$ be an D4571: Real matrix such that

(i)	$A$ is an D5871: Invertible real matrix
(ii)	$B$ is an D2089: Inverse matrix for $A$

We have \begin{equation} A B = B A = I_N \end{equation} Transposing both sides, we have \begin{equation} B^T A^T = A^T B^T = I_N \end{equation} $\square$