# Proof regarding skew symmetric matrices

1. Oct 14, 2009

### nietzsche

1. The problem statement, all variables and given/known data

Show that if A is skew symmetric, then Ak is skew symmetric for any positive odd integer k.

2. Relevant equations

3. The attempt at a solution

Wow, I have no idea how to prove this. I'm guessing there's going to be induction involved. I know that the base case of k = 1 is true, because A1 = A, where A is skew symmetric. But after that, I don't know what to do at all.

Any suggestions on where to go from here?

2. Oct 14, 2009

### Dick

Skew-symmetric means A^(T)=(-A). (T=transpose). Does that help? Can you use that to show A^3 is skew-symmetric?

3. Oct 14, 2009

### nietzsche

\begin{align*} A &= -A^T\\ A^3 &= (-A^T)^3\\ &= (-A^T)(-A^T)(-A^T)\\ &= (A^2)^T(-A^T)\\ &= (-AA^2)^T\\ &= (-A^3)^T \end{align*}

so A3 is skew symmetric.

Thanks again for your help Dick. Now, how can I go about proving this for the rest of the positive odd integers. Will induction work? I'm trying to think of a way to do it, but I'm stuck on the "odd" integers part.

4. Oct 14, 2009

### Dick

Sure, induction will work. But that's probably overkill. (A^n)^T=(A^T)^n. Just factor out the (-1)^n.

5. Oct 14, 2009

### nietzsche

OH...haha thanks very much. I tend to over analyze sometimes...