Gujarat BoardEnglish MediumSTD 12 ScienceMathsMATRICES5 Marks
Question
Prove by Mathematical Induction that$ (A′)^n = (A^n)′,$ where $\text{n}\in\text{N}$ for any square matrix A.
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Answer
Let $P(n) : (A')^n = (A^n)$'$\therefore$ $P(1) : (A') = (A)'$
$\Rightarrow A' = A'$
$\Rightarrow P(1)$ is true.
Now, let $P(k) = (A')^k = (A^k)',$ where $\text{k}\in\text{N}$
and $P(k + 1) : (A')^{K+1} = (A')^kA'$
$= (A^k)'A'$
$= (AA^k)' (as (AB)' = B'A')$
$= (A^{k+1})'$
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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