InterviewSolution
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InterviewSolution
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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 `P(n): (A)^(n)=(A^(n))` `therefore P(1):(A)^(1)=(A)` `rArr A=ArArrP(1)` is true Now, `P(k):(A)^(n)=(A^(k))` where `k in N` and `P(k+1) .(A)^(k+1)=(A^(k+1))` Where `P(k+1)` is true whenever P (k) is true. `therefore P(k+1)(A)^(k),(A)^(1)= [A^(k+1)]` `(A^(k)) ,(A)=[A^(k+1)]` `(A.A^(k))=[A^(k+1)]` `(A^(k+1))=[A^(k+1)]` |
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