If A = begin{bmatrix} cos 2theta & -sin 2thet | vedclass

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

Question

(A) Given $A = \begin{bmatrix} \cos 2	heta & -\sin 2	heta \ \sin 2	heta & \cos 2	heta \end{bmatrix}$.The transpose of $A$ is $A^{T} = \begin{bmat

Vedclass · Accepted Answer

$\frac{\pi}{6}$

Vedclass · Answer

(A) Given $A = \begin{bmatrix} \cos 2	heta & -\sin 2	heta \ \sin 2	heta & \cos 2	heta \end{bmatrix}$.The transpose of $A$ is $A^{T} = \begin{bmatrix} \cos 2	heta & \sin 2	heta \ -\sin 2	heta & \cos 2	heta \end{bmatrix}$.Adding $A$ and $A^{T}$:$A + A^{T} = \begin{bmatrix} \cos 2	heta + \cos 2	heta & -\sin 2	heta + \sin 2	heta \ \sin 2	heta - \sin 2	heta & \cos 2	heta + \cos 2	heta \end{bmatrix} = \begin{bmatrix} 2\cos 2	heta & 0 \ 0 & 2\cos 2	heta \end{bmatrix}$.This can be written as $(2\cos 2	heta)I$, where $I$ is the $2 	imes 2$ identity matrix.Given $A + A^{T} = I$, we have $(2\cos 2	heta)I = I$.Comparing the elements, we get $2\cos 2	heta = 1$, which implies $\cos 2	heta = \frac{1}{2}$.Since $\cos \frac{\pi}{3} = \frac{1}{2}$, we have $2	heta = \frac{\pi}{3}$, which gives $	heta = \frac{\pi}{6}$.

If $A = \begin{bmatrix} \cos 2\theta & -\sin 2\theta \\ \sin 2\theta & \cos 2\theta \end{bmatrix}$ and $A + A^{T} = I$, where $I$ is the $2 \times 2$ identity matrix and $A^{T}$ is the transpose of $A$, then the value of $\theta$ is:

Similar Questions

If $A$ and $B$ are square matrices of the same order,then which of the following properties holds true for the transpose of their product?

If $A = \begin{bmatrix} 3 & \sqrt{3} & 2 \\ 4 & 2 & 0 \end{bmatrix}$ and $B = \begin{bmatrix} 2 & -1 & 2 \\ 1 & 2 & 4 \end{bmatrix}$,verify that $(A')' = A$.

If $A$ is a symmetric matrix and $n \in N$,then $A^n$ is

For the matrix $A = \begin{bmatrix} 1 & 5 \\ 6 & 7 \end{bmatrix}$,verify that $(A - A^{\prime})$ is a skew-symmetric matrix.

If $A = \begin{bmatrix} 3 & x-1 \\ 2x+3 & x+2 \end{bmatrix}$ is a symmetric matrix, then the value of $x$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module