Answer
See below.
Work Step by Step
The characteristic equation of the matrix is
$det(xI_2-A)=
det\left(\begin{bmatrix}
x& -1&-1 \\
-1& x&-1\\
-1&-1&x\\
\end{bmatrix} \right)=0$
Hence $x^3-x^2-3x-1=0\\(x+1)((x-1)^2-2)=0$
Thus the eigenvalues are $x=-1$ with multiplicity $1$, $x=1+\sqrt2$ with multiplicity $1$, and $x=1-\sqrt2$ with multiplicity 1. A is symmetric; thus by Theorem 7.7, the corresponding $x=-1$'s eigenspace will have a dimension of $1$, the corresponding $x=1+\sqrt2$'s eigenspace will have a dimension of $1$ and the corresponding $x=1-\sqrt2$'s eigenspace will have a dimension of $1$.
