Question

Let $P \in \mathbb{M}_4 (\mathbb{R})$ be such that $P^4$ is the zero matrix, but $P^3$ is a nonzero matrix. Then which one of the following is FALSE?

• A. For every nonzero vector $v \in \mathbb{R}^4$, the subset $\\{v, Pv, P^2v, P^3v\\}$ of the real vector space $\mathbb{R}^4$ is linearly independent.
• B. The rank of $P^k$ is $4 - k$ for every $k \in \\{1,2,3,4\\}$.
• C. 0 is an eigenvalue of $P$.
• D. If $Q \in \mathbb{M}_4 (\mathbb{R})$ is such that $Q^4$ is the zero matrix, but $Q^3$ is a nonzero matrix, then there exists a nonsingular matrix $S \in \mathbb{M}_4 (\mathbb{R})$ such that $S^{-1}QS = P$.

Answer 1

Answer: (A)

For every nonzero vector $v \in \mathbb{R}^4$, the subset $\\{v, Pv, P^2v, P^3v\\}$ of the real vector space $\mathbb{R}^4$ is linearly independent.

Answer 2

The correct option is (A): For every nonzero vector $v \in \mathbb{R}^4$, the subset $\\{v, Pv, P^2v, P^3v\\}$ of the real vector space $\mathbb{R}^4$ is linearly independent.