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

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

Answers (1)

Answer: Hence proved,

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.

Posted by

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