Show that $\displaystyle (A ^{-1})^T = (A ^T)^{-1}$

Assuming A is invertible.

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- Sep 9th 2009, 01:31 PMRedKManMatrix Inverse Proof
Show that $\displaystyle (A ^{-1})^T = (A ^T)^{-1}$

Assuming A is invertible. - Sep 9th 2009, 01:59 PMMatt Westwood
If you can take the assumption that the determinant of a matrix and its transpose are equal, you can do it by induction and the definition of the inverse as it is defined in terms of cofactors.

Here's Wikipedia:

Invertible matrix - Wikipedia, the free encyclopedia

... it might help. - Sep 9th 2009, 02:17 PMpedrosorio
$\displaystyle A A^{-1} = I $

$\displaystyle (A A^{-1})^T = (I)^T$

$\displaystyle (A^{-1})^T A^T = I$

By definition of inverse matrix, $\displaystyle (A^{-1})^T$ is the inverse of $\displaystyle A^T$, hence:

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