# Proof

• February 11th 2010, 09:10 PM
kaylakutie
Proof
Not sure how to approach this one.

Prove $(A^{-1})^T = (A^T)^{-1}$.
• February 11th 2010, 09:18 PM
danielomalmsteen
that is equivalent to prove that

$({A}^{-1})^t \cdot {A}^t = I$

Owned by the transpose

$(A \cdot {A}^{-1})^t = I$
• February 11th 2010, 10:17 PM
Roam
Quote:

Originally Posted by kaylakutie
Not sure how to approach this one.

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

Hi there,

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

Keeping in your mind the fact that $I^T = I$ you get

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

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

This completes your proof.