Not sure how to approach this one.

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

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- Feb 11th 2010, 08:10 PMkaylakutieProof
Not sure how to approach this one.

Prove $\displaystyle (A^{-1})^T = (A^T)^{-1}$. - Feb 11th 2010, 08:18 PMdanielomalmsteen
that is equivalent to prove that

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

Owned by the transpose

$\displaystyle (A \cdot {A}^{-1})^t = I$ - Feb 11th 2010, 09:17 PMRoam
Hi there,

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

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

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

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

This completes your proof.