IMC 1999 · Problem 1

a) Show that for any $m \in \mathbb{N}$ there exists a real $m \times m$ matrix $A$ such that $A^3 = A + I$, where $I$ is the $m \times m$ identity matrix.

b) Show that $\det A > 0$ for every real $m \times m$ matrix satisfying $A^3 = A + I$.