CUET PG 2022 — Mathematics PYQ

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Question

CUET PG 2022 — Mathematics PYQ

CUET PG | Mathematics | 2022

If $A = [cos B sin B am p; - sin B am p; cos B]$ then $A + A^{T} = I$ for $B$ equals to

Choose the correct answer:

Explanation

1. Identify the Transpose of $A$

Given the matrix:

A = \begin{bmatrix} \cos B & -\sin B \\ \sin B & \cos B \end{bmatrix}

The transpose $A^T$ is obtained by interchanging the rows and columns:

A^T = \begin{bmatrix} \cos B & \sin B \\ -\sin B & \cos B \end{bmatrix}

2. Calculate $A + A^T$

Now, we add the corresponding elements of both matrices:

A + A^T = \begin{bmatrix} \cos B + \cos B & -\sin B + \sin B \\ \sin B - \sin B & \cos B + \cos B \end{bmatrix}

A + A^T = \begin{bmatrix} 2\cos B & 0 \\ 0 & 2\cos B \end{bmatrix}

3. Compare with Identity Matrix $I$

According to the problem, $A + A^T = I$ . The identity matrix $I$ is:

I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}

Comparing the corresponding elements:

2\cos B = 1

\cos B = \frac{1}{2}

4. Solve for $B$

We know that the cosine of an angle is $\frac{1}{2}$ at $60^\circ$ or $\frac{\pi}{3}$ radians.

General solution:

B = 2n\pi \pm \frac{\pi}{3}, \text{ where } n \in \mathbb{Z}

Final Answer:

The principal value for $B$ is:

$B = \frac{π}{3}$