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

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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