Question

Prove by Mathematical Induction that (A′)ⁿ = (Aⁿ)′, where n ∈ N for any square matrix A.

Answer 1

Let P(n): (A′)ⁿ = (Aⁿ)′

∴ P(1): (A′) = (A)′

⇒ A' = A'

⇒ P(1) is true.

Now, let P(k) = (A')^k = (A^k)'

Where k ∈ N 

And P(k + 1): (A')^k+1 = (A')^kA'

= (A')^k'A'

= (AA^k)'   .....(As (AB)' = B'A')

= (A^k+1)'

Thus P(1) is true and whenever P(k) is true P(k + 1) is true,

So, P(n) is true for all n ∈ N