If a matrix multiplied by its transpose equals the original matrix, is it symmetric? Here's the question:

Prove: If ATA = A, then A is symmetric and A = A2

I tried to solve this by using inference.

  
*
  
*Assume A is symmetric, prove A = A2
  
  
*
  
*If A is symmetric, then by definition ATA = A2. Since ATA = A, then A = A2.
  
  
*Assume A = A2, prove A is symmetric
  
  
*
  
*??
  
  

I wasn't sure what to put for the second part to prove 1. Is an idempotent matrix by definition symmetric? It would make sense, but I couldn't find anything definitive.
Am I going about this correctly, or should I take a different approach?
 A: You seem to have misunderstood what you need to prove. You seem to be attempting to prove that

If $A^TA$, then ($A$ is symmetric if and only if $A=A^2$).

But that is not what you are being asked to prove! What you are being asked to prove is that

If $A^TA=A$, then ($A$ is symmetric and $A=A^2$).

So you are not allowed to just assume that $A$ is symmetric or that $A=A^2$; you need to prove these things from only the hypothesis that $A^TA=A$.
Your only assumption is that $A^TA=A$.  


*

*To prove $A$ is symmetric, remember that $A$ is symmetric if and only if $A^T=A$. But if $A=A^TA$, then $A^T = (A^TA)^T = \cdots$

*To prove that $A=A^2$, argue like you did above, since you have now shown that $A$ is symmetric.
Note that in general, the two statements I wrote above in the grey boxes are not logically equivalent. If the second one holds, then the first one must (because both sides of the "if and only if" will be true whenever the premise is true); but you can have the former one be true and the latter one not be true. For example, "if $x$ is a real number, then ($x$ is positive if and only if $-x$ is negative)" is true, but "if $x$ is a real number, then ($x$ is positive and $-x$ is negative)" is false (as witnessed by $x=-1$). 
