Thread: 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!
Thanks in advance!

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!
Thanks in advance!

I would guess they meant

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

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