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

The Correct Option is A

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.