# Thread: Linear Algebra Proof with invertible matrices

1. ## 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!

2. ## 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 $\displaystyle A^T?$

3. ## 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.

4. ## Re: Linear Algebra Proof with invertible matrices

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.