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

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 $n \in N$.