# Matrix Power

• September 3rd 2008, 06:11 PM
Brokescholar
Matrix Power
If AB = BA and p is a nonnegative integer, prove that (AB)^ p = A^p B^p.

I've been sitting on this one for an hour and I'm just ridiculously confused!
• September 3rd 2008, 06:22 PM
NonCommAlg
Quote:

Originally Posted by Brokescholar
If AB = BA and p is a nonnegative integer, prove that (AB)^ p = A^p B^p.

I've been sitting on this one for an hour and I'm just ridiculously confused!

by induction over p. if p = 1 it's clear. assuming it's true for p, then: $(AB)^{p+1}=A(BA)^pB=A(AB)^pB=A(A^pB^p)B=A^{p+1}B^{ p+1}. \ \ \ \square$
• September 3rd 2008, 07:41 PM
Brokescholar
Awesome! You're a genius thanks for the help!