Prove by Mathematical Induction
Question:

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

Solution:

Let P(n): (A¢)n = (An) ¢

So, P(1): (A¢) = (A) ¢

A¢ = A¢

Hence, P(1) is true.

Now, let P(k) = (A¢)k = (Ak) ¢, where k ∈ N

And,

P(k + 1): (A¢)k+1 = (A¢)k

= (Ak) ¢A¢

= (AAk) ¢

= (Ak+1) ¢

Hence, P(1) is true and whenever P(k) is true P(k + 1) is true.

Therefore, P(n) is true for all n ∈ N.