GATE | GATE IT 2006 | Question 26

What are the eigenvalues of the matrix P given below
 \begin{pmatrix}    a     & 1     & 0\\    1     & a     & 1 \\   0     & 1     &a \end{pmatrix}
(A) a, a -√2, a + √2
(B) a, a, a
(C) 0, a, 2a
(D) -a, 2a, 2a


Answer: (A)

Explanation: det(A-λ.I))=0
(a-λ)*[(a-λ)2-1] -1*(a-λ)= 0
(a-λ)3 – 2(a-λ) = 0
(a-λ)((a-λ)2 – 2) = 0
(a-λ)(a-λ+√2)(a-λ-√2) = 0

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