Question #40381

This question tests the interaction between nilpotent matrices, system consistency, and the point discontinuity of the signum function.

The trap lies in the determinant $\det(M(x)) = -(\text{sgn}^2(x) - 1)^2$.

Students often incorrectly assume $\text{sgn}^2(x) = 1$ for all real $x$, concluding that the matrix is always singular ($\det = 0$) and nilpotent.

Consequently, they only check the consistency conditions for $x \neq 0$, finding $x=1$ and $x=4$, but completely miss the singularity trap at $x=0$.

At $x=0$, $\text{sgn}(0)=0$, the rank fluctuates, and the matrix regains full rank ($\det = -1$), guaranteeing a unique solution and adding $x=0$ to the solution set.