If A invertible, is $A + A^{T}$ invertible ?

Question

I'm trying to solve the following problem about matrices:

If $A$ is invertible is $A + A^{T}$ invertible?

This is what I have done so far:

$A + A^{T}$

$A(A^{-1}) + A^{T}(A^{T})^{-1} = 2I$

$I + A^{T}(A^{T})^{-1} = 2I$

$A^{T}(A^{T})^{-1} = I$

I believe that at this point I have to stop right? The answer is that it's not invertible but does this prove it?

Answer 1

I don't see how your argument shows that $A + A^T$ is not invertible - it's certainly not generally true that $A$ being invertible implies that $A + A^T$ is not.

All you need, however, is a specific counterexample. Why not try considering, e.g.

$$A = \left(\begin{array}{cc} 0 & 1 \\ -1 & 0\end{array} \right)$$