**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¢)kA¢

= (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.