Project ForgeConsider 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?
- AOnly I and III are necessarily true
- BOnly II is necessarily true
- COnly I and II are necessarily true
- Only II and III are necessarily true
Explanation
If P had two distinct eigenvalues, it would have two independent eigenvector directions, contradicting the premise that ALL eigenvectors are multiples of a single vector. So P must have a single repeated eigenvalue with a geometric multiplicity of only 1 — statement II is necessarily true. Since the eigenspace (dimension 1) is smaller than the algebraic multiplicity of the repeated eigenvalue (at least 2, for an n≥2 matrix), P is defective and cannot be diagonalized — statement III is necessarily true. But statement I doesn't follow: the repeated eigenvalue could be nonzero (e.g. a Jordan block with eigenvalue 5), in which case P would still be invertible. So only II and III are necessarily true.
Source: GATE CS 2018 (2018)
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