Linear Algebra Theorem

[Steven G]

Theorem: If Matrix A is skew symmetric then A^K is skew symmetric for any positive integer K.

The above is a theorem in my Linear Algebra book.

However I have reasons to think it is wrong. Can someone please point out the error in my counterexample.

1st I will tell you what I know. For A to be skew symmetric we must have A^T= -A. Also (AB)^T = B^T*A^T

Suppose A is skew symmetric. Then (A²)^T=(AA)^T= A^TA^T (-A)(-A) = A² and not -A²

 

[Steven G]

This theorem was asked to be proven in the homework section so it may be untrue.
The more I think about it the more I am positive that it is not true.
Can I please have one more person verify my suspicion.
Thanks.

 

[HallsofIvy]

Let \(\displaystyle T= \begin{bmatrix}0 & -1 \\ 1 & 0\end{bmatrix}\).
Then \(\displaystyle T^2= \begin{bmatrix}-1 & 0 \\ 0 & -1\end{bmatrix}\)