Question

The exhaustive set of characteristic roots of an idempotent matrix are

• A. $\{0,1,-1\}$
• B. $\{0,1\}$
• C. $\left\{0\right\}$
• D. $\varphi$

Answer 1

The correct option is B $\{0,1\}$

Since A is an idempotent matrix, we have
$A^2=A$
Let X be a latent vector of the matrix A corresponding to the latent root $\lambda$ so that
$AX=\lambda X$ (1)
$(A-\lambda I)X=O$
such that $X\ne O$
On pre-multiplying Equation (1) by A, we get
$A\left(AX\right)=A\left(\lambda X\right)=\lambda \left(AX\right)$
$\left(AA\right)X=\lambda \left(AX\right)$
$AX={\lambda }^{2}X$ ($\because A^2=A$)
$\lambda X={\lambda }^{2}X$ $(\because AX=\lambda X)$
$({\lambda }^2-\lambda )X=O$
${\lambda }^2-\lambda =0$ $(\because X\ne 0)$
$\lambda (\lambda -1)=0$
$\Rightarrow \lambda =0,\lambda =1$