Find $Tr(B)+Tr(C)$

Question

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?

Answer 1

The value of $tr(B)+tr(C)$ may not be unique. But if it really is, then we can obtain it by any solution, e.g., by this "trivial" solution:

$$ \begin{pmatrix} -1 & 1 \cr 0 & 2\end{pmatrix}= \begin{pmatrix} -1 & 1 \cr 0 & 1\end{pmatrix}^3+ \begin{pmatrix} 0 & 0 \cr 0 & 1\end{pmatrix}^3 $$ So $tr(B)+tr(C)=0+1=1$.

Answer 2

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 $$B=A+I\;\;\;,\;\;\;C=-I$$