IMC 1995 · Day 1 · P5

We have that $$(A + tB)^n = A^n + t P_1 + t^2 P_2 + \cdots + t^{n-1} P_{n-1} + t^n B^n$$ for some matrices $P_1, P_2, \dots, P_{n-1}$ not depending on $t$.

Assume that $a, p_1, p_2, \dots, p_{n-1}, b$ are the $(i,j)$-th entries of the corresponding matrices $A^n, P_1, P_2, \dots, P_{n-1}, B^n$. Then the polynomial $$b t^n + p_{n-1} t^{n-1} + \cdots + p_2 t^2 + p_1 t + a$$ has at least $n+1$ roots $t_1, t_2, \dots, t_{n+1}$. Hence all its coefficients vanish. Therefore $A^n = 0$, $B^n = 0$, $P_i = 0$; and $A$ and $B$ are nilpotent.