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  4. If A = [ cos θ sin θ - sin θ cos θ] then lim n arrow ∞ (1/n) An is

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Q. If $A = \begin{bmatrix}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{bmatrix}$ then $\lim _{n \rightarrow \infty} \frac{1}{n} A^{n}$ is

Matrices

Solution:

$A^{n}= \begin{bmatrix}\cos n \theta & \sin n \theta \\ -\sin n \theta & \cos n \theta\end{bmatrix}$
$\Rightarrow \frac{1}{n} A^{n}= \begin{bmatrix}\frac{\cos n \theta}{n} & \frac{\sin n \theta}{n} \\ -\frac{\sin n \theta}{n} & \frac{\cos n \theta}{n}\end{bmatrix}$
But $-1 \leq \cos n \theta \leq 1$ and $-1 \leq \sin n \theta \leq 1$
$\displaystyle \lim _{n \rightarrow \infty} \frac{\sin n \theta}{n}=0, \displaystyle \lim _{n \rightarrow \infty} \frac{\cos n \theta}{n}=0$
$\Rightarrow \displaystyle \lim _{n \rightarrow \infty} \frac{1}{n} A^{n}= \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}$