Question

If $A$ is an idempotent matrix satisfying $(I-0.4A=(I-\alpha A)$ (where I is the unit matrix of same order as that of $A,A$ is not a null matrix), then $\left|\frac{2}{\alpha }\right|$ is

Answer 1

Answer: 3

$A^2=A$
$(I-0.4A=(I-\alpha A)$
$I=\left(I-0.4A\right)\left(I-\alpha A\right)$
$I=I-\alpha A-0.4A+0.4\alpha A^2$
$0.4\alpha A-0.4A-\alpha A=0$
$A\left(0.4\alpha -0.4-\alpha \right)=0$
$\therefore 0.4\alpha -\alpha =0.4$
$-0.6\alpha =0.4$
$\alpha =\frac{-2}{3}$
$\Rightarrow \frac{2}{\alpha }=-3\Rightarrow \left|\frac{2}{\alpha }\right|=3$