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

The Correct Option is A, B, C

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