Let A be an nxn invertible matrix.

I can't figure out how to prove the following:

t(A^-1) = (t(A))^-1. So the transpose of A inverse is equal to the inverse of A transpose.

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- Sep 14th 2009, 12:04 PMgrandunificationproof involving inverses of invertible matrices
Let A be an nxn invertible matrix.

I can't figure out how to prove the following:

t(A^-1) = (t(A))^-1. So the transpose of A inverse is equal to the inverse of A transpose. - Sep 14th 2009, 12:20 PMcourteous
So, if I understand, you wonder if equality $\displaystyle (A^{-1})^T=(A^T)^{-1}$ holds.

Multiply both sides by $\displaystyle A^T$: $\displaystyle (A^{-1})^TA^T=(A^T)^{-1}A^T$.

On the right-hand side of the equation you have identity matrix: $\displaystyle (A^{-1})^TA^T=I$.

And left-side, $\displaystyle (A^{-1})^TA^T$, equals identity matrix $\displaystyle (A(A^{-1}))^T=I$.

(Cool) - Sep 14th 2009, 12:29 PMgrandunification
Ya, I just worked it out right before I checked your answer. Thanks anyways though.