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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?

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This is not true in general, but there might be some interesting classes of matrices for which this holds (excluding the obvious symmetric ones). A good candidate could be, e.g., a class of positive real matrices, i.e., matrices satisfying $x^TAx>0$ for all nonzero $x$ (note $A$ does not need to be symmetric). – Algebraic Pavel Sep 20 '13 at 11:29
up vote 3 down vote accepted

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.

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

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