If A = begin{bmatrix} cos theta & -sin theta | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Given $A = \begin{bmatrix} \cos 	heta & -\sin 	heta \ \sin 	heta & \cos 	heta \end{bmatrix}$.By the property of rotation matrices, $A^n = \be

Vedclass · Accepted Answer

$\begin{bmatrix} \cos 2 	heta & -\sin 2 	heta \ \sin 2 	heta & \cos 2 	heta \end{bmatrix}$

Vedclass · Answer

(A) Given $A = \begin{bmatrix} \cos 	heta & -\sin 	heta \ \sin 	heta & \cos 	heta \end{bmatrix}$.By the property of rotation matrices, $A^n = \begin{bmatrix} \cos n 	heta & -\sin n 	heta \ \sin n 	heta & \cos n 	heta \end{bmatrix}$.For $n = 100$, we have $A^{100} = \begin{bmatrix} \cos 100 	heta & -\sin 100 	heta \ \sin 100 	heta & \cos 100 	heta \end{bmatrix}$.Given $	heta = \frac{2 \pi}{7}$, so $100 	heta = \frac{200 \pi}{7}$.We can write $\frac{200 \pi}{7} = \frac{196 \pi + 4 \pi}{7} = 28 \pi + \frac{4 \pi}{7}$.Since $\cos(28 \pi + \alpha) = \cos \alpha$ and $\sin(28 \pi + \alpha) = \sin \alpha$, we get $A^{100} = \begin{bmatrix} \cos \frac{4 \pi}{7} & -\sin \frac{4 \pi}{7} \ \sin \frac{4 \pi}{7} & \cos \frac{4 \pi}{7} \end{bmatrix} = A^2$.Comparing this with the options, none of the provided options match $A^2$ exactly. However, evaluating the expression, $A^{100} = A^2$ is the correct result.

If $A = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}$ and $\theta = \frac{2 \pi}{7}$, then $A^{100} = A \times A \times \dots \times A$ ($100$ times) is equal to:

Similar Questions

$A$ square matrix $[a_{ij}]$ in which $a_{ij} = 0$ for $i \neq j$ and $a_{ij} = k$ (constant) for $i = j$ is called a

If $X = \begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix}$,then the value of $X^n$ is

If $A = \begin{bmatrix} 1 & -1 & 1 \\ 2 & -1 & 0 \\ 1 & 0 & 0 \end{bmatrix}$, then $A^5 =$

If $A^2 = A$ is a square matrix such that $(I - A)^n = I - A$ for $n \geq 1$,then what is the value of $(I + A)^2 - 3A$?

Given $3\begin{bmatrix} x & y \\ z & w \end{bmatrix} = \begin{bmatrix} x & 6 \\ -1 & 2w \end{bmatrix} + \begin{bmatrix} 4 & x+y \\ z+w & 3 \end{bmatrix}$,find the values of $x, y, z$ and $w$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module