Prove Idempotency of A: Homework Statement

[jmarzouq]

Homework Statement

Prove that if (A^τ)A = A, then A is idempotent. [Hint: First show that (A^τ)A = A^τ]

Homework Equations

N/A

The Attempt at a Solution

I've gotten to the hint portion by taking the transpose of both sides, but have been unable to get that far past that. I've tried right side multiplying by A^-1 and have gotten this far:
A^τ = A^τ(A^-1), then, taking the transpose of each side yields
A = [(A^-1)^τ]A

I can't figure out how to get rid of the transpose/inverse from there. Any help would be greatly appreciated. Thanks!

 

[jbunniii]

You can't take the inverse because you don't know that A is invertible. Indeed, if A is invertible and idempotent, then A must be the identity matrix. (Can you prove that?)

You are given that $A^T A = A$, and you have shown that $A^T A = A^T$. The left hand sides of these two equations are the same, and therefore the right hand sides must also be the same. What does that imply about A?

 

[jmarzouq]

Ah, I see now. So A is equal to its own transpose so you can just substitute A back into the initial equation for A^T and get A^2 = A proving it's idempotent. I really appreciate the help!