Question

Let \(A=\left(\begin{array}{ccc}1 & 0 & 0 \\ 0 & \text { cost } & \sin t \\ 0 & -\sin t & \cos t\end{array}\right)\)
et \(\lambda_{1}, \lambda_{2}, \lambda_{3}\) be the roots of \(\operatorname{det}\left(A-\lambda I_{3}\right)=0\), where \(I_{3}\) denotes the identity matrix. If \(\lambda_{1}+\lambda_{2}+\lambda_{3}=\sqrt{2}+1\), then the set f possible values of \(t,-\pi \leq t<\pi\) is
(A) a void set
(B) \(\left\{\frac{\pi}{4}\right\}\)
(C) \(\left\{-\frac{\pi}{4}, \frac{\pi}{4}\right\}\)
(D) \(\left\{-\frac{\pi}{3}, \frac{\pi}{3}\right\}\)

Answer 1

Ans: (C)
$$
\text { Hint : } \begin{aligned}
\left|\begin{array}{ccc}
1-\lambda & 0 & 0 \\
0 & \cos t-\lambda & \sin t \\
0 & -\sin t & \cos t-\lambda
\end{array}\right|=0 \\
\Rightarrow(1-\lambda)(\cos t-\lambda)^{2}+\sin ^{2} t=0 \\
\Rightarrow(1-\lambda)\left(\lambda^{2}-2 \lambda \cos t+\cos ^{2} t\right)+\sin ^{2} t=0 \\
\Rightarrow \lambda^{2}-2 x \cos t+\cos ^{2} t-\lambda^{3}+2 \lambda^{2} \cos t+\lambda \cos ^{2} t+\sin ^{2} t=0 \\
\Rightarrow-\lambda^{3}+\lambda^{2}(1+2 \cos t)+\lambda\left(\cos ^{2} t-2 \cos t\right)+1=0
\end{aligned}
$$

$$
\begin{aligned}
&\lambda_{1}+\lambda_{2}+\lambda_{3}=1+2 \cos t=1+\sqrt{2} \\
&\therefore \cos t=\frac{1}{\sqrt{2}} \Rightarrow t=\frac{\pi}{4},-\frac{\pi}{4}
\end{aligned}
$$

Answer 2

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Answer 3

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