GO Classes CS/DA 2025 | Weekly Quiz 5 | Linear Algebra | Question: 6

Answer 1

Option B

if \( A \) is a nilpotent matrix with \( k = 2 \) (meaning \( A^2 = 0 \)), then \( A \) cannot have a non-zero eigenvalue. 

To understand why, let's suppose that \( A \) has a non-zero eigenvalue \( \lambda \). Then there exists a non-zero vector \( v \) such that \( Av = \lambda v \). 

Now, let's consider \( A^2v = A(Av) = A(\lambda v) = \lambda(Av) = \lambda(\lambda v) = \lambda^2v \).

Since \( A^2 = 0 \), we have \( \lambda^2v = 0 \). But since \( v \) is non-zero, this implies that \( \lambda^2 = 0 \). However, if \( \lambda^2 = 0 \), then \( \lambda = 0 \). Thus, any eigenvalue of \( A \) must be zero if \( A \) is nilpotent with \( k = 2 \).

Answer 2

• A. Distinct eigenvalues have LI eigenvectors but same eigenvalue can also have LI eigenvectors.
  
  
• B. $A^2 = 0$ then $A$ has only $0$ as its eigenvalue.
  
  
• C. True
  
  
• D. True