Matrices and Determinants
Eigenvalues and Eigenvectors
Premium Question
Grade 12
Question:
If $A$ is an idempotent matrix, i.e. $A^2 = A$, then every eigenvalue $\lambda$ of $A$ must satisfy:
(1) $\lambda = 1$ only
(2) $\lambda^2 = \lambda$, so $\lambda \in \{0, 1\}$
(3) $\lambda = 0$ only
(4) $\lambda^2 = 1$, so $\lambda \in \{-1, 1\}$
Step-by-Step Solution
Key Concept: If $Av = \lambda v$ for an eigenvector $v \neq 0$, apply $A$ again to both sides to express $A^2 v$ in two different ways, and use $A^2 = A$ to obtain an equation purely in $\lambda$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (2)