homework - Show a matrix as a combination of other matrices and long division The topic we are dealing with here is polynomial division.
The question is:
We are given a polynomial: $f(x) = (x+1)(x-1)^2$, and a matrix $D \in R^{nxn}$ such that $f(D)=0$
Using only $I, D, D^2$ find:
1) $D^5+3D^2-D+I$
2) $D^{2013}$
3) $D^{-1}$
What I did: For question (1) I divided the polynomial $x^5 +3x^2-x+1$ by $x^3-x^2-x+1 = (x+1)(x-1)^2$ and I got:
$x^5 +3x^2-x+1 = (x^2+x+2)*(x^3-x^2-x+1)+5x^2-1$ but $(x^3-x^2-x+1)=$ when $x=D$ and so:
$D^5+3D^2-D+I = 5D^2-I$
So that solves question (1). But question (2)...I have no clue.

Hint: Suppose $x^{2013} = (x+1)(x-1)^2q(x) + r(x)$ with $\operatorname{deg}(r)\le2$. Then $r(1)=1,\ r(-1)=-1$ and $r'(1)=2013$.
