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.