Find $Tr(B)+Tr(C)$ If $B,C$ are $2 \times 2 $ matrices with integer entries such that:
$$\begin{bmatrix}
-1 &1 \\ 
 0& 2
\end{bmatrix}=B^3+C^3$$
Find value of $Tr(B)+Tr(C)$
My try:
Taking trace on both sides we get:
$$Tr(B^3)+Tr(C^3)=1$$
Any clue from here?

Let $\displaystyle A=\begin{bmatrix}
-1 &1 \\ 
 0& -2
\end{bmatrix}$ and $\displaystyle I =\begin{bmatrix}1&0 \\0&1\end{bmatrix}$.
Then Using Characteristic Equation(Caly -Hamilton Theorem)
$|A-\lambda I|=0\Rightarrow \begin{vmatrix}
-1-\lambda &&  1& \\ 
 0&  & -2-\lambda\\ 
 \end{vmatrix}=0$
$\Rightarrow \lambda^2+3\lambda+2=0\Rightarrow A^2+3A+2I=O$
$$ \Rightarrow A^3+3A^2+2A=O$$
$\Rightarrow A=A^3+3A^2+3A+I^3-I^3$
$$\Rightarrow A=(A+I)^3-I^3=B^3+C^3$$
So we have $$B=A+I\;\;\;,\;\;\;C=-I$$
