Problem
If the column vector $\left[\begin{array}{c}5 \\ -5\end{array}\right]$ is an eigenvector of the $2 \times 2$ matrix $A=\left[\begin{array}{cc}4 & a \\ -1 & 1\end{array}\right] \cdots$ corresponding to the eigenvalue $\lambda$, then $a+\hat{\lambda}=?$
Solution
Step 1 ：Solve the equation Av = λv, where v = \(\begin{bmatrix} 5 \\ -5 \end{bmatrix}\) and A = \(\begin{bmatrix} 4 & a \\ -1 & 1 \end{bmatrix}\)
Step 2 ：Substitute the solution a = 2 and λ = 2 into the expression a + λ
Step 3 ：Calculate the final answer: \(\boxed{4}\)
