If the column vector $\left[\begin{array}{l}-2 \\ -2\end{array}\right]$ is an eigenvector of the $2 \times 2$ matrix $A=\left[\begin{array}{ll}a & 1 \\ 1 & 2\end{array}\right]$ corresponding to the eigenvalue $\lambda$, then $a+\lambda=$ ? (A) 7 (B) 4 (C) 8 (D) 5 (E) 6

Step 1 ：Given eigenvector \(v = \begin{bmatrix} -2 \\ -2 \end{bmatrix}\) and matrix \(A = \begin{bmatrix} a & 1 \\ 1 & 2 \end{bmatrix}\), we need to find \(a\) and \(\lambda\) that satisfy the equation \(Av = \lambda v\).

Step 2 ：Solving the equation, we get \(a = 2\) and \(\lambda = 3\). Therefore, \(a + \lambda = 2 + 3 = \boxed{5}\).