Browse Questions
Home  >>  CBSE XII  >>  Math  >>  Matrices

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

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.