Question
Prove by Mathematical Induction that (A′)n = (An)′, where $\text{n}\in\text{N}$ for any square matrix A.
✓
Answer
Let P(n) : (A')n = (An)'
$\therefore$
P(1) : (A') = (A)'⇒ A' = A'
⇒ P(1) is true.
Now, let P(k) = (A')k = (Ak)', where
$\text{k}\in\text{N}$and P(k + 1) : (A')K+1 = (A')kA'
= (Ak)'A'
= (AAk)' (as (AB)' = B'A')
= (Ak+1)'
Thus P(1) is true and whenever P(k) is true P(k + 1) is true.
So, P(n) is true for all $\text{n}\in\text{N}.$
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