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Provide solution for rd sharma math class 12 chapter Algebra of matrices exercise 4.3 question 63

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Answer: Hence proved,\left(A^{T}\right)^{n}=\left(A^{n}\right)^{T}  for all n \in N.

Hint: We use the principle of mathematical induction.

Given: A is a square matrix.

Prove: \left(A^{T}\right)^{n}=\left(A^{n}\right)^{T}  for all n \in N.                 

Let  P(n) = \left(A^{T}\right)^{n}=\left(A^{n}\right)^{T}  for all n \in N.                     …(i)

Step 1: put n=1 in eqn(i)

\left(A^{T}\right)^{\mathbf{1}}=\left(A^{1}\right)^{T} \quad\left[x^{\mathbf{1}}=x\right]            …(ii)

A^{T}=A^{T}

Thus, P(n) is true for n=1

Assume that P(n) is true for n \in N

P(k)=\left(A^{T}\right)^{k}=\left(A^{k}\right)^{T}                                 …(iii)

To prove that P(k+1) is true, we have

\left(A^{T}\right)^{k+1} =\left(A^{T}\right)^{k}\left(A^{T}\right)^{1}

                    \\\\=\left(A^{k}\right)^{T}\left(A^{1}\right)^{T} \quad \ \ \ \text { [ using (ii) and (iii) } ]\\\\ =\left(A^{k+1}\right)^{T}

Thus, P(k+1) is true, whenever P(k) is true.

Hence, by principle of mathematical induction P(n) is true for all n \in N.

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