For the matrix A = [[2, 1], [0, 2]], which of the following is true regarding its geometric and algebraic multiplicities for eigenvalue λ = 2?
Answer and explanation
Correct answer: A. Algebraic multiplicity = 2, Geometric multiplicity = 1
- Step 1: Characteristic polynomial: det(A - λI) = (2 - λ)^2 = 0. The root λ = 2 has algebraic multiplicity 2.
- Step 2: Eigenspace is null space of (A - 2I) = [[0, 1], [0, 0]]. The equation is y = 0, giving only 1 linearly independent eigenvector [1, 0]^T.
- Step 3: Geometric multiplicity = nullity(A - 2I) = 1.
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