Question

Consider the matrix A = [[50,70],[70,80]] whose eigenvectors corresponding to eigenvalues λ₁ and λ₂ are x₁ = [70, λ₁−50] and x₂ = [λ₂−80, 70], respectively. The value of x₁ᵀx₂ is __________.

Accepted Answer

The correct answer is 0 (± 0.1). A is symmetric (off-diagonal entries both 70). For a real symmetric matrix, eigenvectors corresponding to distinct eigenvalues are always orthogonal — so x₁ᵀx₂ = 0 regardless of the specific eigenvalue values, without needing to compute λ₁, λ₂ explicitly.

Explanation