What are the eigenvalues of the matrix P given below [Tex] \begin{pmatrix} a &…

Let λ be an eigenvalue of P. Then:

P - λI =
[Tex]\begin{pmatrix}
a-\lambda & 1 & 0 \\
1 & a-\lambda & 1 \\
0 & 1 & a-\lambda
\end{pmatrix}[/Tex]

So,

[Tex]\det(P-\lambda I) = (a-\lambda)\left((a-\lambda)^2 - 1\right) - (a-\lambda)[/Tex]

[Tex]= (a-\lambda)\left((a-\lambda)^2 - 2\right)[/Tex]

For eigenvalues, det(P - λI) = 0. Therefore:

[Tex](a-\lambda)\left((a-\lambda)^2 - 2\right)=0[/Tex]

This gives:
λ = a, a - √2, a + √2.

Hence, the correct option is a, a - √2, a + √2.