# Linear Algebra Proof with invertible matrices

• Mar 13th 2013, 07:38 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, 08:16 PM
Gusbob
Re: Linear Algebra Proof with invertible matrices
We know \$\displaystyle A\$ is invertible, and so \$\displaystyle A^{-1}A=I\$

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

What can you then conclude about the invertibility of \$\displaystyle A^T?\$
• Mar 14th 2013, 12:49 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, 02:24 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, \$\displaystyle A^{-1}A= I\$, not \$\displaystyle (A^T)^{-1}\$.

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