**Step 1: **

Let P(n) be (A′)^{n}=(A^{n})′

Let P(n) be true for all n∈ N

For n=1 (A′)^{1}=(A^{1})′

⇒(A)′=A′.

Hence LHS = RHS.

Hence P(n) is true for n = 1

**Step 2: **

Let P(n) be true for n=k.

Put n = k

(A′)^{k}=(A^{k})′

Multiply A′ on both the side

(A′)^{k}.A′=(A^{k})′.A′

⇒ A^{′k}.A′ = (A^{k})′.A′

⇒ A′^{k}.A′^{1 }= (A^{k}.A^{1})′

⇒ (A′)^{k+1 }= (A^{k+1})′

Hence P(n) is true for n = k+1.