Question

If $f\left(x\right)=x^3-x^2-2x,$ then possible number of $2\times 2$ matrices that can be formed by taking the elements which are roots of the equation $f\left(|x|\right)=0$ is

Answer 1

We have,
$f\left(x\right)=x^3-x^2-2x=x(x-2)(x+1)$
$\therefore f\left(x\right)=0$ has roots $x=0,x=-1,x=2$
$\therefore f\left(|x|\right)=0$ has roots $|x|=0$ and $|x|=2$ $\{\because |x|=-1\text{is not possible}\}$
$\therefore$ Elements of the matrix are $0,\pm 2$
Each element can be filled in $3$ ways
Hence, Possible number of $2\times 2$ matrices $=3^4=81$