Hello,
Can you please help solve the following problem:
Given two matrices A,B and AB=BA, I need to prove that for every n:
$\displaystyle {(AB)}^{n}={A}^{n}{B}^{n}$.
Thanks in advance,
Michael
I have tried to prove it by induction:
1) I checked for n=1:
$\displaystyle {(AB)}^{1}=(A)}^{1}(b)}^{1}$
2)I assumed for n=k it is true:
$\displaystyle {(AB)}^{k}=(A)}^{k}(b)}^{k}$
3)I checked for n=k+1:
$\displaystyle {(AB)}^{k+1}=(A)}^{k+1}(b)}^{k+1}$
$\displaystyle {(AB)}\cdot {(AB)}^{k}={A}\cdot {A}^{k}\cdot {B}\cdot {B}^{k}$
Then, after switching places and using the induction assumption , I reached:
$\displaystyle {(AB)}\cdot {(AB)}^{k}={(AB)}\cdot {(AB)}^{k}$
is this ok?
I think the last couple steps are... out of order or something.
You have
$\displaystyle (AB)^{k+1}=(AB)(AB)^k=ABA^kB^k=AA^kBB^k=A^{k+1}B^{ k+1}$.
Your last line is $\displaystyle {(AB)}\cdot {(AB)}^{k}={(AB)}\cdot {(AB)}^{k}$. But... this is clearly true, it doesn't say anything about.. anything.
No, it doesn't. You need to prove that $\displaystyle (AB)^{k+1}= (AB)(AB)^k= A^{k+1}B^{k+1}$. The trivial observation that $\displaystyle (AB)(AB)^k= (AB)(AB)^k$ doesn't do t
Having said that $\displaystyle (AB)^k= A^kB^k$, you can conclude that $\displaystyle (AB)^{k+1}= (AB)(AB)^k= (AB)(A^kB^k)$. Now, what can you conclude from that?
It looks as though you need to use induction twice if you really want to pin this result down rigorously.
First, use induction to show that $\displaystyle AB^n = B^nA$ for every $\displaystyle n\geqslant1.$
Then use that result to prove by induction on m that $\displaystyle A^mB^n = B^nA^m$ for every $\displaystyle m\geqslant1$ (and hence in particular when m=n).