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If A and B are square matrices of the sa...

If A and B are square matrices of the same order such that `A B = B A`, then proveby induction that `A B^n=B^n A`. Further, prove that `(A B)^n=A^n B^n`for all `n in N`.

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We are given, `AB = BA`
We have to prove, `AB^n = B^nA`
When `n = 1`,
`AB^1 = B^1A => AB = BA`
So, given equation is true for `n =1`
Let this equation is true for `n=k`.
Then, `AB^k = B^kA`
Now, we have to prove it is true for `n = k+1` that is `AB^(k+1) = B^(k+1)A`
...
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