= Solution
A <fixed point> obeys $(I-M)X=c$. Thus the necessary and sufficient existence condition is
$$
\boxed{c\in\operatorname{range}(I-M).}
$$
If $1$ is not an <eigenvalue> of $M$, then $I-M$ is an <invertible matrix> and the unique constant solution is $\boxed{X=(I-M)^{-1}c}$. If $I-M$ is singular but the equation is consistent, all constant solutions are $X_0+\ker(I-M)$ for any particular solution $X_0$. For an existing <fixed point>, $x_{k+1}-X=M(x_k-X)$; the successive difference in the question likewise obeys $\delta x_{k+1}=M\delta x_k$.
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