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 $A = \begin{bmatrix} \cos 	heta & -\sin 	heta \ \sin 	heta & \cos 	heta \end{bmatrix}$.Using the property of rotation matrices, $A^n =

Vedclass · Accepted Answer

$\begin{bmatrix} 0 & 1 \ -1 & 0 \end{bmatrix}$

Vedclass · Answer

(A) Given $A = \begin{bmatrix} \cos 	heta & -\sin 	heta \ \sin 	heta & \cos 	heta \end{bmatrix}$.Using the property of rotation matrices, $A^n = \begin{bmatrix} \cos(n	heta) & -\sin(n	heta) \ \sin(n	heta) & \cos(n	heta) \end{bmatrix}$.Thus, $A^{-3} = \begin{bmatrix} \cos(-3	heta) & -\sin(-3	heta) \ \sin(-3	heta) & \cos(-3	heta) \end{bmatrix} = \begin{bmatrix} \cos(3	heta) & \sin(3	heta) \ -\sin(3	heta) & \cos(3	heta) \end{bmatrix}$.Given $	heta = \pi/6$, then $3	heta = 3(\pi/6) = \pi/2$.Substituting the value, $A^{-3} = \begin{bmatrix} \cos(\pi/2) & \sin(\pi/2) \ -\sin(\pi/2) & \cos(\pi/2) \end{bmatrix} = \begin{bmatrix} 0 & 1 \ -1 & 0 \end{bmatrix}$.

If $A = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}$, then the matrix $A^{-3}$ when $\theta = \pi/6$ is equal to...

Similar Questions

Let $A$ be a square matrix of order $3$. Choose the correct option regarding the following statements:
$I$. There exists a matrix $B$ of order $3$ such that $AB = I_3$
$II$. There exists a matrix $C$ of order $3$ such that $CA = I_3$
$III$. $A$ is invertible

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

If $A = \begin{bmatrix} 1 & -1 \\ 2 & 3 \end{bmatrix}$,then $\text{adj } A$ is equal to

If $P = \begin{vmatrix} 1 & \alpha & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4 \end{vmatrix}$ is the adjoint of a $3 \times 3$ matrix $A$ and $\det(A) = 4$, then $\alpha$ is equal to

Let $A = \begin{bmatrix} 3 & 7 \\ 2 & 5 \end{bmatrix}$ and $B = \begin{bmatrix} 6 & 8 \\ 7 & 9 \end{bmatrix}$. Verify that $(AB)^{-1} = B^{-1} A^{-1}$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module