pls solve asap

pls solve asap If P and Q are equal matrices of same order such that PQ = QP, then prove by induction that PQ z CP. Further, show that (PQ)n x

Dear student,
let us consider matrix P=A and matrix Q=B
given
A and B are equal 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
hence PQⁿ=QⁿP  for all n belongs to N

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.
hence(PQ)ⁿ=QⁿP^n
regards,