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

Answer

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Answer

Solving the equation, we get $a = 2$ and $λ = 3$ . Therefore, $a + λ = 2 + 3 = 5$ .

Steps

Step 1 ：Given eigenvector $v = [\begin{matrix} - 2 \\ - 2 \end{matrix}]$ and matrix $A = [\begin{matrix} a & 1 \\ 1 & 2 \end{matrix}]$ , we need to find $a$ and $λ$ that satisfy the equation $A v = λ v$ .

Step 2 ：Solving the equation, we get $a = 2$ and $λ = 3$ . Therefore, $a + λ = 2 + 3 = 5$ .

If the column vector [−2−2] is an eigenvector of the 2×2 matrix A=[a112] corresponding to the eigenvalue λ, then a+λ= ?(A) 7(B) 4(C) 8(D) 5(E) 6

Answer

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