Consistent or Inconsistent system

[Baron]

Prove or disprove:

If the system A(Ax) = b is consistent, then Ax = b is consistent. (where x and b are vectors and A is some matrix)

My work:

I'm not sure whether the statement is true or false. I'm leaning towards true because I can't come up with any counterexamples but I having difficulties proving the statement is true.

I tried direct proof, but I don't know whether A is invertible.
I tried to prove by contrapositive by assuming Ax = b is inconsistent and trying to prove A(Ax) = b is inconsistent. My argument is that since Ax = b is inconsistent for any x, A(Ax) = b must also be inconsistent because Ax can be seen as an element of x. (if that makes any sense)

Does this work? Is there a way to prove this by direct proof?

 

[HallsofIvy]

You do not know, directly, whether or not A is invertible, but you do know, because \(\displaystyle A(Ax)= A^2x= b\) is consistent, that \(\displaystyle A^2\) is invertible so what does that tell you about A?

(That is for b non-zero. If b= 0, the equation \(\displaystyle A^2x= 0\) can be consistent without A being invertible. Do that case separately.)