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Let $A = [\begin{array}{ccc} 1 & 0 & 0 \\ 0 & \cos t & \sin t \\ 0 & - \sin t & \cos t \end{array}] .$ Let $λ_{1}, λ_{2}, λ_{3}$ be the roots of det $(A - λ I_{3}) = 0$ where $I_{3}$ denotes the identity matrix. If $λ_{1} + λ_{2} + λ_{3} = \sqrt{2} + 1,$ then the possible value(s) of t, $- π \leq t \leq π$ is

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$\frac{π}{3}$

$- \frac{π}{4}$

$\frac{π}{4}$

$- \frac{π}{3}$

answer is A.

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Detailed Solution

$\det (A - λ I_{3}) = 0$

$|\begin{array}{ccc} 1 - λ & 0 & 0 \\ 0 & \cos t - λ & \sin t \\ 0 & - \sin t & \cos t - λ \end{array}| = 0$

$λ^{3} - λ^{2} (1 + 2 \cos t) + λ (1 + 2 \cos t) - 1 = 0$

$λ_{1} + λ_{2} + λ_{3} = 1 + 2 \cos t$

$1 + 2 \cos t = 1 + \sqrt{2}$

$\cos t = \frac{1}{\sqrt{2}} \Rightarrow t = \pm \frac{π}{4}$

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