Let $F(n, l, i, j)$ be the cardinality of the set \begin{eqnarray*} \{(k_1, \cdots, k_n)\in\mathbb{Z}^{\oplus n}|0\leq k_r\leq l-1\text{ for }1\leq r\leq n\text{, }k_1+\cdots+k_n=lj-i\}. \end{eqnarray*} Define an $n\times n$ matrix $M$ by \begin{eqnarray*} M_{ij}=(-1)^{i+j}F(n, l, i, j). \end{eqnarray*} In fact $M$ is related to the Adams operations on $U(n)$, and I can show using algebraic topology that the eigenvalues are $1, l, l^2, \cdots, l^{n-1}$.

Question: Are there more elementary ways to show this?