Prove Invertibility of Square Matrix A & ATA

[iamsmooth]

Homework Statement

Prove that asquare matrix A is invertible if nad only if A^TA is invertible

Homework Equations

Hmm, can't think of any. A A^-1 = I maybe?

The Attempt at a Solution

I have trouble with these theory questions, so I'm not sure how to approach this.

If something is invertible, that means it's determinant is not 0? So does it have something to do with that?

Thanks

 

[Hurkyl]

If you aren't good at "theory" questions, then try rephrasing them as "calculation" questions!

If something is invertible, that means it's determinant is not 0?

And vice versa as well, right? If the determinant is not zero, then it's invertible. So we have an identity:

{X is invertible} = {det X is not zero}​

What happens if you substitute this into the original question? (There are two things to substitute! Do both!) Is it something that you can solve by doing two calculations?

 

[iamsmooth]

A square matrix A is invertible if and only if A^TA is invertible

Well, If A is invertible, then det(A) is not 0.

det(A) = det(A^T)

The poduct of two invertible matrices is invertible

So therefore A^TA is invertible if and only if A is invertible.

This doesn't seem like proof, nor does it seem coherent :( I suck

 

[VeeEight]

You have the right idea but try to show both directions separately.
If A is invertible, then det(A) is non zero. So what can you say about det(AA^T)?
Similarly, if det(AA^T) is non zero, what can you say about det(A)?
(Also remember that det(AB) = det(A)det(B))