Find $0\neq B,C \in S$ such that $BC=0$ with multiplication and addition of matrix $S=\{a_0I+a_1A+a_2A^2:a_o,a_1,a_2\in \mathbb{Q}\}$

Question

I have the following question :

Find $0\neq B,C \in S$ such that $BC=0$ with multiplication and addition of matrices $S=\{a_0I+a_1A+a_2A^2:a_o,a_1,a_2\in \mathbb{Q}\}$

Hint : $A^3+2A^2-I=0$ (The hint if part of the question)

$A= \left( \begin{array}{ccc} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & -2 \end{array} \right) $

$I= \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right) $

I know from a theorem that the decomposition should be $(B^2+C+D)(E^2+F+G)$ meaning lower than 3 since the order of the original polynom is $3$, But I have no idea how to find it.

Any idea?

Answer 1

Using that $x^3+2x^2-1=(x+1)(x^2+x-1)$:

$$0=A^3+2A^2-I=(A+I)(A^2+A-I),$$ $$0=A^3+2A^2-I=(A^2+A-I)(A+I).$$ Then, $B=A+I\in S,$ $C=A^2+A-I\in S$ with $B\ne 0$, $C\ne 0$ and $BC=CB=0.$