I have a set of $A_{n\times n}$ matrices that satisfies $I+A+A^2+A^3+A^4 = 0$.
I can see that $A^5 = I \Rightarrow A^{k+5}=A^k$. How is this possible if A doesn't consists of a series of permutation matrices?
The homework question is: what could be said about $\dim(\operatorname{span}(I,A,A^2,A^3,....))$?