2. Let $A \in \mathcal{M}_{2}(\mathbb{R})$ that satisfies the relation $A^{2}+A+I_{2}=O_{2}$.

a) Find a matrix that satisfies the relation given in the statement.

b) Calculate the determinant of the matrix $A$.
2. a) Consider the matrix $A=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)$ with $a+d=-1$ and $ad-bc=1$, for example

$A=\left(\begin{array}{cc}-1 & -1 \\ 1 & 0\end{array}\right)$
b) $A^{2}+A+I_{2}=O_{2}$ Multiplying the relation by $A-I_{2}$ results in $A^{3}-I_{2}=O_{2}$, hence $A^{3}=I_{2}$ and $\operatorname{det}\left(A^{3}\right)=1$, i.e., $\left.\begin{array}{c}\operatorname{det} A)^{3}=1 \\ \operatorname{det} A \in \mathbb{R}\end{array}\right\} \Rightarrow \operatorname{det} A=1$. $4 p$