Math Help - If A is an invertible matrix, then A+A^T is skew-symmetric. (Proof).

1. If A is an invertible matrix, then A+A^T is skew-symmetric. (Proof).

My teacher wrote:

(A+A^T)^T = A^T + (A^T)^T = A^T + A

While I get this algebraically, I don't see how it proves skew-symmetry. Like how does this relate to A = -A^T?! Also, I just wanted to confirm if the reason why the stuff (=bolded part) in (stuff)^T is A+A^T because we want to force them to be square matrices since you can only add matrices that have the same dimensions.

Any input would be greatly appreciated!

2. Originally Posted by s3a
My teacher wrote:

(A+A^T)^T = A^T + (A^T)^T = A^T + A

While I get this algebraically, I don't see how it proves skew-symmetry. Like how does this relate to A = -A^T?! Also, I just wanted to confirm if the reason why the stuff (=bolded part) in (stuff)^T is A+A^T because we want to force them to be square matrices since you can only add matrices that have the same dimensions.

Any input would be greatly appreciated!
I would guess they meant $\left(A-A^{\top}\right)^{\top}=A^{\top}-A=-\left(A-A^{\top}\right)$

3. So the way it should have been phrased was?: "If A is an invertible matrix, then A-A^T is skew-symmetric."

4. Originally Posted by s3a
So the way it should have been phrased was?: "If A is an invertible matrix, then A-A^T is skew-symmetric."
That's my guess