# Linear Algebra Proof with invertible matrices

• Mar 13th 2013, 08:38 PM
widenerl194
Linear Algebra Proof with invertible matrices

Let A be an n x n invertible matrix. Prove that AT is invertible and determine its inverse in terms of A-1.

Any hints or suggestions would be appreciated!
• Mar 13th 2013, 09:16 PM
Gusbob
Re: Linear Algebra Proof with invertible matrices
We know $A$ is invertible, and so $A^{-1}A=I$

Taking the transpose of both sides gives $A^T(A^{-1})^T=I$

What can you then conclude about the invertibility of $A^T?$
• Mar 14th 2013, 01:49 PM
widenerl194
Re: Linear Algebra Proof with invertible matrices
So then can I just say, since these two are equal A transpose is invertible. Then it's inverse is A inverse times A.
• Mar 14th 2013, 03:24 PM
HallsofIvy
Re: Linear Algebra Proof with invertible matrices
Quote:

Originally Posted by widenerl194
So then can I just say, since these two are equal A transpose is invertible. Then it's inverse is A inverse times A.

No, $A^{-1}A= I$, not $(A^T)^{-1}$.

What you can say is that the inverse of A transpose is the transpose of A inverse.