Prove that if $AA^T=A$ then $A^3=A$

Question

The approach I'd like to use to prove this particular property necessitates that $A$ be invertible, but I don't wish to assume this (though it would certainly make the task simpler).

Is there some property which shows $A$ to be invertible which I am overlooking, or is it that perhaps I need to use a different method of proof?

Answer 1

From the given:

$$A = AA^T= (A^T)^TA^T= (AA^T)^T = A^T$$

$ \implies A = A^T$

Now reuse this in the initial expression to get:

$$A = AA^T = AA = A^2$$

$\implies A = A^2$

Thus

$$A^3 = A^2 A = AA = A^2 = A$$

Answer 2

Invertibility has nothing to do with this. Note that $$ A^T=(AA^T)^T=AA^T=A. $$ Then $A^2=AA^T=A $.