The solution for the matrix system $(A-X)X=0$

Question

$A$ and $X$ are all matrices. Also, $X\geq0$ element-wise and $A\ne X$. Is $X=0$ the only solution for this nonlinear system $(A-X)X=0$?

Answer 1

What about $A=I_2$ and $X=\pmatrix{1&0\\ 0&0}$?

Answer 2

$(a - x)x = 0$ in the reals (aka $1\times 1$ matrices) has solution $a = x$ or $x= 0$, and since you can send the real numbers to $n\times n$ matrices homomorphically, $x \to {\rm diag}_n(x)$, there exists an infinite number of example matrices such that the only solutions are $A = X$ or $X = 0$. That's all I can say for now - got to go to work.