Consider a matrix P whose only eigenvectors are the multiples of [1, 4]^T. Consider the following statements. I. P does not have an inverse II. P has a repeated eigenvalue III. P cannot be diagonalized Which one of the following options is correct? A. Only I and III are necessarily true B. Only II is necessarily true C. Only I and II are necessarily true D. Only II and III are necessarily true

GATE 2018 · Engineering Mathematics · Eigen Value · medium

Answer: D. Only II and III are necessarily true

1. Analyze Statement I: P does not have an inverse: A repeated nonzero eigenvalue (e.g., lambda = 3 with multiplicity 2) would give P invertible but still have only one eigenvector direction. So P CAN be invertible.
2. Analyze Statement II: P has a repeated eigenvalue: If P (assumed 2x2 or larger) had distinct eigenvalues, each distinct eigenvalue contributes at least one independent eigenvector. Two distinct eigenvalues on a 2x2 matrix yield two independent eigenvectors, contradicting that only [1,4]^T direction exists. Hence eigenvalue must be repeated.
3. Analyze Statement III: P cannot be diagonalized: With only one eigenvector direction, the geometric multiplicity is 1. For a 2x2 matrix with repeated eigenvalue (algebraic multiplicity 2), geometric multiplicity 1 < algebraic multiplicity 2. So P cannot be diagonalized.
4. Conclusion: Only II and III are necessarily true. Statement I fails when eigenvalue is nonzero and repeated.