Question

If $A=\left(\begin{array}{cc}cos\theta & isin\theta \\ isin\theta & cos\theta \end{array}\right)$, where $i=\sqrt{-1}$, then by principle of Mathematical Induction then $A^n=\left[\begin{array}{cc}cosn\theta & isinn\theta \\ isinn\theta & cosn\theta \end{array}\right]$.

• A. True
• B. False

Answer 1

The correct option is A True

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

To Check whether the given statement is true

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

Put $n=1$

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

So,$A^n$ is true for $n=1$

Let $A^n$ is true for $n=k$, so

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

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

Now,$A^{k+1}=A^k\times A$

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

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

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

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

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 integer.

Hence the given statement is true.