Question

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

Answer 1

$A^{2}=A$
$( I -0.4 A =( I -\alpha A )$
$I=\left(I - 0 .4 A\right)\left(I - \alpha A\right)$
$I=I-\alpha A-0.4A+0.4\alpha A^{2}$
$0.4\alpha A-0.4A-\alpha A=0$
$A\left(0 . 4 \alpha - 0 .4 - \alpha \right)=0$
$\therefore 0.4\alpha -\alpha =0.4$
$ \, -0.6\alpha =0.4$
$ \, \alpha =\frac{- 2}{3}$
$\Rightarrow \frac{2}{\alpha }=-3\Rightarrow \left|\frac{2}{\alpha }\right|=3$