  Given $n \times n$ matrices, $A_1, \dots, A_k$, consider the linear operator
$L(A_1,\dots,A_k) \, \colon \; \operatorname{M}_n \to \operatorname{M}_n$ given
by \[ L(A_1,\dots,A_k)(A_{k+1})= \sum_{\sigma\in S_{k+1}}
\operatorname{sign}(\sigma) A_{\sigma(1)}A_{\sigma(2)} \cdots A_{\sigma(k+1)}.
\] The Amitsur-Levitzki theorem asserts that $L(A_1, \ldots, A_k)$ is
identically $0$ for every $k \geq 2n-1$. Dixon and Pressman conjectured that if
$k$ is an even number between $2$ and $2n - 2$, then the kernel of $L(A_1,
\ldots, A_k)$ is of dimension $k$ for $A_1, \ldots, A_k\in
\operatorname{M}_n(\mathbb{R})$ in general position. We prove this conjecture
using graph-theoretic techniques.
