Chapter 7 - Eigenvalues and Eigenvectors - 7.3 Symmetric Matrices and Orthogonal Diagonalization - 7.3 Exercises - Page 370: 11

See below.

The characteristic equation of the matrix is $det(xI_2-A)= det\left(\begin{bmatrix} x-2& -1 \\ -1& x-2\\ \end{bmatrix} \right)=0$ Hence $(x-2)^2-1=x^2-4x+4-1=0\\(x-3)(x-1)=0$ Thus the eigenvalues are $x=3,1$ both with multiplicity $1$. A is symmetric; thus by Theorem 7.7, the corresponding eigenspaces will also have a dimension of $1$.