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

To solve this problem, let's analyze each statement regarding the matrix \( P \in \mathbb{M}_4 (\mathbb{R}) \) with the properties that \( P^4 \) is the zero matrix and \( P^3 \) is not the zero matrix.

1. Statement: 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.
   • A set of vectors \(\{v, Pv, P^2v, P^3v\}\) will be linearly independent if the only solution to the equation \(c_0v + c_1Pv + c_2P^2v + c_3P^3v = 0\) is \(c_0 = c_1 = c_2 = c_3 = 0\).
   • Since \(P^4 = 0\), applying \(P\) four times to any vector results in the zero vector. Thus, \(P^3v\) can be expressed as a linear combination of the previous vectors for some vectors \(v\), implying that the set cannot be linearly independent for all nonzero vectors \(v\).
   • Therefore, this statement is FALSE.
2. Statement: The rank of \( P^k \) is \( 4 - k \) for every \( k \in \{1,2,3,4\} \).
   • Since \( P^4 = 0 \) and \( P^3 \neq 0 \), it follows that the nilpotency index of \( P \) is 4. Thus, the rank-nullity theorem can be used to infer the rank of each power of \( P \).
   • For \( P^k \), the column space dimension drops by 1 with each further multiplication, due to the nature of nilpotent matrices, matching the given ranks: \(4 - k\).
   • Thus, this statement is TRUE.
3. Statement: 0 is an eigenvalue of \( P \).
   • A matrix is nilpotent if and only if 0 is its only eigenvalue. Since \(P\) is nilpotent (as \(P^4 = 0\)), its only eigenvalue is indeed 0.
   • Hence, this statement is TRUE.
4. Statement: 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 \( S^{-1}QS = P \).
   • Nilpotent matrices of the same size with the same index of nilpotency are similar. Therefore, \(P\) and \(Q\) must be similar as both are nilpotent of index 4.
   • This implies the existence of a nonsingular matrix \(S\) such that \(S^{-1}QS = P\).
   • Thus, this statement is TRUE.

Therefore, the false statement is: 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.