# Idempotent Proof

## Homework Statement

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

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 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?