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## 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!