Proof regarding skew symmetric matrices

[nietzsche]

Homework Statement

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

Homework Equations

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?

 

[Dick]

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

 

[nietzsche]

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

$$\begin{align*}<br /> A &= -A^T\\<br /> A^3 &= (-A^T)^3\\<br /> &= (-A^T)(-A^T)(-A^T)\\<br /> &= (A^2)^T(-A^T)\\<br /> &= (-AA^2)^T\\<br /> &= (-A^3)^T<br /> \end{align*}$$

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.

 

[Dick]

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

 

[nietzsche]

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

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