IMC 2003 · Day 1 · P3

The minimal polynomial of $A$ is a divisor of $3 x^3 - x^2 - x - 1$. This polynomial has three different roots. This implies that $A$ is diagonalizable: $A = C^{-1} D C$ where $D$ is a diagonal matrix. The eigenvalues of the matrices $A$ and $D$ are all roots of polynomial $3 x^3 - x^2 - x - 1$. One of the three roots is 1, the remaining two roots have smaller absolute value than 1. Hence, the diagonal elements of $D^k$, which are the $k$th powers of the eigenvalues, tend to either 0 or 1 and the limit $M = \lim D^k$ is idempotent. Then $\lim A^k = C^{-1} M C$ is idempotent as well.