Question

If for the matrix \( A = \begin{bmatrix} \tan x & 1 \\ -1 & \tan x \end{bmatrix} \), \( A + A' = 2\sqrt{3}I \), then the value of \( x \in \left[ 0, \frac{\pi}{2} \right] \) is:

• A. \( 0 \)
• B. \( \frac{\pi}{4} \)
• C. \( \frac{\pi}{3} \)
• D. \( \frac{\pi}{6} \)

Hint:

For matrix equations involving transpose, simplify the addition or subtraction to identify key terms and compare with the identity matrix.

Answer 1

The Correct Option is C

Step 1: {Expand the matrix equation}
The given matrix \( A = \begin{bmatrix} \tan x & 1 \\ -1 & \tan x \end{bmatrix} \). The transpose is: \[ A' = \begin{bmatrix} \tan x & -1 \\ 1 & \tan x \end{bmatrix}. \] Adding \( A \) and \( A' \): \[ A + A' = \begin{bmatrix} \tan x & 1 \\ -1 & \tan x \end{bmatrix} + \begin{bmatrix} \tan x & -1 \\ 1 & \tan x \end{bmatrix} = \begin{bmatrix} 2\tan x & 0 \\ 0 & 2\tan x \end{bmatrix}. \] Step 2: {Compare with the given equation}
The given equation is: \[ A + A' = 2\sqrt{3}I, \] where \( I \) is the identity matrix. Thus: \[ \begin{bmatrix} 2\tan x & 0 \\ 0 & 2\tan x \end{bmatrix} = \begin{bmatrix} 2\sqrt{3} & 0 \\ 0 & 2\sqrt{3} \end{bmatrix}. \] Step 3: {Solve for \( \tan x \)}
Equating elements, we get: \[ 2\tan x = 2\sqrt{3} \quad \Rightarrow \quad \tan x = \sqrt{3}. \] Thus, \( x = \frac{\pi}{3} \) (in the interval \( \left[ 0, \frac{\pi}{2} \right] \)). 
Conclusion: The value of \( x \) is \( \frac{\pi}{3} \).