Question

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

Answer 1

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

For n = 1, we have:

Therefore, the result is true for n = 1.

Let the result be true for n = k.

Now, we prove that the result is true for n = k + 1.

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

Thus, by the principle of mathematical induction, we have

Now, we prove that for all n ∈ N

For n = 1, we have:

Therefore, the result is true for n = 1.

Let the result be true for n = k.

Now, we prove that the result is true for n = k + 1.

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

Thus, by the principle of mathematical induction, we have, for all natural numbers.