Let $k$ be a field, let $n$ be a positive integer, and let $A=(a_{ij})_{1 \leq i,j \leq n} \in M_n(k)$. Define the [characteristic polynomial](/page/Characteristic%20Polynomial) of $A$ by
For each subset $S \subset \{1,\dots,n\}$, let $A_S$ denote the principal submatrix of $A$ whose rows and columns are indexed by $S$ in increasing order, with the convention that $A_\varnothing$ is the empty matrix and $\det A_\varnothing=1$. Then, for every $r \in \{1,\dots,n\}$,