Question

If A and B are square matrices of the same order such that AB = BA, then prove by induction that AB" = B"A. Further, prove that (AB)" = A"B" for all n ∈ N

Answer 1

A and B are square matrices of the same order such that AB = BA.

To prove  P(n) : AB" = B"A, `n in N`

For n = 1, we have:

Therefore, the result is true for n = 1.

Let the result be true for n = k.

Therefore, the result is true for n = k + 1.

Thus, by the principle of mathematical induction, we have AB" = B"A, `n in N`

Now, we prove that (AB)" = A"B" for all n ∈ N

For n = 1, we have:

Therefore, the result is true for n = k + 1.

Thus, by the principle of mathematical induction, we have (AB)" = A"B", for all natural numbers.