Let $M$ be a non-zero $3\times 3$ matrix satisfying $M^3=0$, where $0$ is the $3\times 3$ zero matrix. Then
$(A)\det(\frac{1}{2}M^2+M+I)\neq0$
$(B)\det(\frac{1}{2}M^2-M+I)=0$
$(C)\det(\frac{1}{2}M^2+M+I)=0$
$(D)\det(\frac{1}{2}M^2-M+I)\neq0$
This is a multiple correct choice type question.More than one may be correct answers.
My attempt:
Let $M$ be a non-singular matrix,therefore its inverse exists.
$M^3=0$
Multiply both sides by $M^{-1}$ to get,$M^2=0$.
Again multiply both sides by $M^{-1}$ to get,$M=0$
But this is a contradiction.So $M$ is a singular matrix.Its inverse does not exist.
But i dont know how to solve further.Please help me.Thanks.