Quick Linear Algebra Proof: Why is A^T nonsingular if A is nonsingular?

[squaremeplz]

Homework Statement

Prove that if A is nonsingular then A^T is nonsingular and

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

Homework Equations

(AB)^T = (B^T)(A^T)

The Attempt at a Solution

Step 1: Multiply both sides by B^T

B^T * (A^T)^(-1) = B^T * (A^(-1))^T

B^T * (A^T)^(-1) = (A^(-1)*B)^T

(A^(-1)*B)^T = (A^(-1)*B)^T

I feel that my last step is wrong. Any suggestions would be helpful.

I was also noticing that if I take just the right side of the equation and do this

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

= I^T = I

which suggests that the left side has to equal I if I multiply it by A^T as well so

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

which makes sense just looking at it.

Thanks!

 

[Dick]

Start from A*A^(-1)=I. Now take the transpose of both sides and use your relevant equation. Stare at it for a while and figure out what it means.