Question

Let \[ P = \begin{pmatrix} 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{pmatrix} \in M_5(\mathbb{C}). \]
Which of the following statements is/are TRUE?

• A. \( \text{nullity}(P - I) \geq 2 \), where \( I \) is the \( 5 \times 5 \) identity matrix.
• B. \( P \) has 4 distinct eigenvalues in \( \mathbb{C} \).
• C. \( P \) has 3 distinct eigenvalues in \( \mathbb{R} \).
• D. If \( \lambda \) is an eigenvalue of \( P \), then there exists a positive integer \( n \) such that \( \lambda^n = 1 \).

Hint:

When analyzing eigenvalues of a matrix, check the determinant and trace of the matrix, as they give important information about the sum and product of eigenvalues.

Answer 1

The Correct Option is A, D

Step 1: Eigenvalues of \( P \).
The matrix \( P \) is a \( 5 \times 5 \) matrix that has eigenvalues \( \pm 1 \) and \( 0 \), with multiplicities that need to be computed. By inspecting \( P \), we find that it has 3 distinct eigenvalues: \( 1, -1, 0 \). Thus, option (C) is true.

Step 2: Analyze option (A).
The nullity of \( P - I \) is the number of linearly independent solutions to \( (P - I) \mathbf{v} = 0 \). Since \( P \) has eigenvalue \( 1 \), this is not necessarily true for all cases. Thus, option (A) is false.

Step 3: Analyze option (B).
The matrix \( P \) has only 3 distinct eigenvalues: \( 1, -1, 0 \), so option (B) is false.

Step 4: Analyze option (D).
Since \( 1 \) and \( -1 \) are eigenvalues of \( P \), they satisfy \( \lambda^2 = 1 \). Hence, option (D) is true.

Step 5: Conclusion.
The correct answer is (C), as \( P \) has 3 distinct eigenvalues in \( \mathbb{R} \).