Question

Let \(A=\left(\begin{array}{lll}3 & 0 & 3 \\ 0 & 3 & 0 \\ 3 & 0 & 3\end{array}\right) .\) Then the roots of the equation \(\operatorname{det}\left(A-\lambda I_{3}\right)=0\) (where \(I_{3}\) is the identity matrix of order 3 ) are
(A) \(3,0,3\)
(B) \(0,3,6\)
(C) \(1,0,-6\)
(D) \(3,3,6\)

Answer 1

Ans: (B)
$$
\text { Hint : Let } \begin{aligned}
\left(A-\lambda I_{3}\right)=0 \quad \Rightarrow\left|\begin{array}{ccc}
3-\lambda & 0 & 3 \\
0 & 3-\lambda & 0 \\
3 & 0 & 3-\lambda
\end{array}\right|=0 \\
\Rightarrow(3-\lambda)^{3}-9(3-\lambda)=0 \Rightarrow(3-\lambda)\left[(3-\lambda)^{2}-3^{2}\right]=0 \\
\Rightarrow 3-\lambda=0 \text { or } 3-\lambda-3=0 \text { or } 3-\lambda+3=0 \quad \Rightarrow \lambda=0,3 \text { or } 6
\end{aligned}
$$

Answer 2

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Answer 3

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