Prove by Mathematical Induction that $(A')^n=(A^n)'$,where $n\in N$ for any square matrix A.

Answer 1

Toolbox:

• We use the principle of mathematical induction, where we need to prove P(n) is true for n=1, n=k, n=k+1

Step1:

Let P(n) be $(A')^n=(A^n)'$

Let P(n) be true for all $n\in N$

For n=1

$(A')^1=(A^1)'$

$\Rightarrow (A)'=A'.$

Hence LHS=RHS.

Hence P(n) is true for n=1

Step2:

Let P(n) be true for n=k.

Put n=k

$(A')^k=(A^k)'$

Multiply $A'$ on both the side

$(A')^k.A'=(A^k)'.A'$

$\Rightarrow {A'^k}.A'=(A^k)'.A'$

$\Rightarrow {A'^k}.A'^1=(A^k.A^1)'$

$\Rightarrow (A')^{k+1}=(A^{k+1})'$

Hence P(n) is true for n=k+1.