Chapter 7 - Eigenvalues and Eigenvectors - 7.3 Symmetric Matrices and Orthogonal Diagonalization - 7.3 Exercises - Page 370: 12

See below.

The characteristic equation of the matrix is $det(xI_2-A)= det\left(\begin{bmatrix} x-2& 0 \\ 0& x-2\\ \end{bmatrix} \right)=0$ Hence $(x-2)^2=0$ Thus the eigenvalue is $x=2$ with multiplicity $2$. A is symmetric; thus by Theorem 7.7, the $x=2$'s corresponding eigenspace will have a dimension of $2$.