Matrices and Determinants
Eigenvalues and Eigenvectors
Premium Question
Grade 12
Question:
If $A$ is a nilpotent matrix, i.e. $A^k = O$ for some positive integer $k$, then every eigenvalue of $A$ equals:
(1) $0$
(2) $1$
(3) $-1$
(4) $k$
Step-by-Step Solution
Key Concept: If $Av = \lambda v$ with $v \neq 0$, repeatedly apply $A$ to obtain $A^k v = \lambda^k v$. Now use the condition $A^k = O$ to see what this forces $\lambda^k$, and hence $\lambda$, to be.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)