Question

For α, β ∈ R , α ≠ β, if −2 and 5 are the eigenvalues of the matrix 𝑀=[1−𝛼1+𝛽𝛽 𝛼+𝛽] and 𝑋 = [𝑥1 𝑥2 ] is an eigenvector of 𝑀 associated to −2, then$M=\begin{bmatrix} 1-α& 1+β\\\ β & a+B& \end{bmatrix}$
and 𝑋 = $\begin{bmatrix} X_1\\\ X_2 \end{bmatrix}$ is an eigenvector of 𝑀 associated to −2, then

• A. 2𝑥1 + 𝑥2 = 0
• B. 𝛽 − 𝛼 = 5
• C. 𝛼 2 − 𝛽 2 = 5
• D. 𝑥1 + 3𝑥2 = 0

Answer 1

Answer: (A), (B), (C)

2𝑥1 + 𝑥2 = 0

Answer 2

The correct options is (A): 2𝑥1 + 𝑥2 = 0, (B): 𝛽 − 𝛼 = 5 and (C): 𝛼 2 − 𝛽 2 = 5