(C)

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Let $\lambda_1, \lambda_2, \lambda_3$ denote the eigenvalues of the matrix

$$A \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos t & \sin t \\ 0 & – \sin t & \cos t \end{pmatrix}.$$ If $\lambda_1+\lambda_2+\lambda_3 = \sqrt{2}+1$, then the set of possible values of $t, \: – \pi \leq t < \pi$, is

- Empty set
- $\{ \frac{\pi}{4} \}$
- $\{ – \frac{\pi}{4}, \frac{\pi}{4} \}$
- $\{ – \frac{\pi}{3}, \frac{\pi}{3} \}$

1 vote

We know that $T_{r}\left ( A \right )$ $=$ $Trace\left ( A \right )$ $=$ sum of principle diagonal elements $=$ sum of eigen values of the matrix.

$\therefore$ $2$$cost$ $+$ $1$ $=$ $\sqrt{2}$ $+$ $1$

$\Rightarrow$ $cost$ $=$ $\frac{1}{\sqrt{2}}$

$\Rightarrow$ $t$ $=$ $-$ $\frac{\pi }{4}$ or $\frac{\pi }{4}$ (for the interval $\left [ -\pi ,\pi \right ]$ , $cosx$ is positive between $\left [ -\frac{\pi }{2},\frac{\pi}{2} \right ]$).

$\therefore$ Option C is the correct answer.

$\therefore$ $2$$cost$ $+$ $1$ $=$ $\sqrt{2}$ $+$ $1$

$\Rightarrow$ $cost$ $=$ $\frac{1}{\sqrt{2}}$

$\Rightarrow$ $t$ $=$ $-$ $\frac{\pi }{4}$ or $\frac{\pi }{4}$ (for the interval $\left [ -\pi ,\pi \right ]$ , $cosx$ is positive between $\left [ -\frac{\pi }{2},\frac{\pi}{2} \right ]$).

$\therefore$ Option C is the correct answer.