#### Please solve RD Sharma class 12 chapter Algebra of matrices exercise 4.3 question 58 maths textbook solution

$A^{n}=\left[\begin{array}{cc}\cos n \theta & i \sin n \theta \\ i \sin n \theta & \cos n \theta\end{array}\right]$ for all $n \in N$

Hint: We use the principle of mathematical induction.

Given:

$A=\left[\begin{array}{cc}\cos \theta & i \sin \theta \\ i \sin \theta & \cos \theta\end{array}\right]$

To show that:

$A^{n}=\left[\begin{array}{cc}\cos n \theta & i \sin n \theta \\ i \sin n \theta & \cos n \theta\end{array}\right]$ for all $n \in N$                  …(i)

Solution:

step 1: Put n=1 in eqn (i)

$A^{1}=\left[\begin{array}{cc}\cos \theta & i \sin \theta \\ i \sin \theta & \cos \theta\end{array}\right]$

So,$A^n$ is true for n=1

Let, $A^n$ is true for n=k, so

$A^{k}=\left[\begin{array}{cc}\cos k \theta & i \sin k \theta \\ i \sin k \theta & \cos k \theta\end{array}\right]$   ... (ii)

Now, we have to show that

$A^{k+1}=\left[\begin{array}{cc}\cos (k+1) \theta & i \sin (k+1) \theta \\ i \sin (k+1) \theta & \cos (k+1) \theta\end{array}\right]$

Now,

$=\left[\begin{array}{cc} \cos k \theta & i \sin k \theta \\ i \sin k \theta & \cos k \theta \end{array}\right]\left[\begin{array}{cc} \cos \theta & i \sin \theta \\ i \sin \theta & \cos \theta \end{array}\right]$

$=\left[\begin{array}{cc} \cos k \theta \cos \theta+i^{2} \sin k \theta \sin \theta & i \cos k \theta \sin \theta+i \sin k \theta \cos \theta \\ i \sin k \theta \cos \theta+i \cos k \theta \sin \theta & i^{2} \sin k \theta \sin \theta+\cos \theta \cos k \theta \end{array}\right] \\\\$

$=\left[\begin{array}{cc} \cos (k+1) \theta & i \sin (k+1) \theta \\ i \sin (k+1) \theta & \cos (k+1) \theta \end{array}\right]$

So, $A^n$ is true for n=k+1 whenever it is true for n=k

Hence, by principle of mathematical induction, $A^n$ is true for all positive integers n.