Gujarat BoardEnglish MediumSTD 12 ScienceMathsMATRICES5 Marks
Question
If A is a square matrix, using mathematical induction prove that $(A^T)^n = (A^n)^T$ for all $n ∈ N$.
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Answer
Let the given statement $P(n)$, be given as
$P(n):\left(A^{\top}\right)^n=\left(A^n\right)^{\top} \text { for all } n \in N \text {. }$
We observe that
$P(1):\left(A^{\top}\right)^1=A^{\top}=\left(A^1\right)^{\top}$
Thus, $\mathrm{P}(\mathrm{n})$ is true for $\mathrm{n}=1$.
Assume that $P(n)$ is true for $n=k \in N$.
$\text { i.e., } P(k):\left(A^{\top}\right)^k=\left(A^k\right)^{\top}$
To prove that $P(k+1)$ is true, we have
$\left(A^T\right)^{k+1}=\left(A^{\top}\right)^k \cdot\left(A^T\right)^1$
$=\left(A^k\right)^{\top} \cdot\left(A^1\right)^{\top}$
$=\left(A^{k+1}\right)^{\top}$
Thus, $P(k+1)$ is true, whenever $P(k)$ is true.
Hence, by the Principle of mathematical induction, $P(n)$ is true for all $n \in N$.
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