Question

If `A^T` is invertible, is A invertible? What about `A^TA`?

Answer 1

Yes and yes

Explanation:

Suppose `A^T` has inverse `(A^T)^(-1)`

For any square matrices `A` and `B`, `A^T B^T = (BA)^T`

Then:

`((A^T)^(-1))^T A = ((A^T)^(-1))^T (A^T)^T=(A^T (A^T)^(-1))^T = I^T = I`

And:

`A ((A^T)^(-1))^T = (A^T)^T ((A^T)^(-1))^T=((A^T)^(-1) A^T)^T = I^T = I`

So `((A^T)^(-1))^T` satisfies the definition of an inverse of `A`.

Then we find:

`(A^T A) (A^-1 (A^T)^(-1)) = A^T (A A^-1) (A^T)^(-1)`

`=A^T I (A^T)^(-1) = A^T (A^T)^(-1) = I`

And:

`(A^-1 (A^T)^(-1)) (A^T A) = A^(-1)((A^T)^(-1) A^T)A`

`=A^(-1) I A = A^(-1)A = I`

So `(A^-1 (A^T)^(-1))` satisfies the definition of an inverse of `A^T A`