7. (20%)

Find the eigenvalues of the following matrix $A$ and determine a basis for each eigenspace. Use the characteristic polynomials and Gauss elimination to obtain your answers.

$$A = \begin{bmatrix} -2 & -6 & -6 \\ -3 & 2 & -2 \\ 3 & 2 & 6 \end{bmatrix}$$