If A is a square matrix, using mathematical induction prove that (A^T)ⁿ = (Aⁿ)^T for all n ∈ ℕ.

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Solution

Let the given statement P(n), be given as
P(n): (A^T)ⁿ = (Aⁿ)^T for all n ∈ ℕ.

We observe that
P(1): (A^T)¹ = A^T = (A¹)^T
Thus, P(n) is true for n = 1.

Assume that P(n) is true for n = k ∈ ℕ.
i.e., P(k): (A^T)^k = (A^k)^T

To prove that P(k + 1) is true, we have
(A^T)^k^{+ 1} = (A^T)^k.(A^T)¹
= (A^k)^T.(A¹)^T
= (A^k^{+ 1})^T
Thus, P(k + 1) is true, whenever P(k) is true.

Hence, by the Principle of mathematical induction, P(n) is true for all n ∈ ℕ.

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