If A is a square matrix, using mathematical induction prove that … - Vidyadip

Question

If $A$ is a square matrix, using mathematical induction prove that $(A^T)^n = (A^n)^T$ for all $n \in N.$

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Answer

Let the given statement $P(n),$ be given as
$P(n): (A^T)^n = (A^n)^T$ for all $n \in N.$
We observe that
$P(1): (A^T)^1 = A^T = (A^1)^T$
Thus, $P(n)$ is true for $n = 1.$
Assume that $P(n)$ is true for $n = k \in N.$
i.e., $P(k): (A^T)^k = (A^k)^T$
To prove that $P(k + 1)$ is true, we have
$(A^T)^{k + 1} = (A^T)^k.(A^T)^1$
$= (A^k)^T.(A^1)^T$
$= (A^{k + 1})^T$
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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