Question

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

 

Answer 1

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.