\begin{definition}[Definition:Minor of Determinant/Notation]
Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$.
Let $D := \map \det {\mathbf A}$ denote the determinant of $\mathbf A$.
Let:
:$\set {a_1, a_2, \ldots, a_k}$ be the indices of the $k$ selected rows of $\mathbf A$
:$\set {b_1, b_2, \ldots, b_k}$ be the indices of the $k$ selected columns of $\mathbf A$
where all of $a_1, \ldots, a_k$ and all of $b_1, \ldots, b_k$ are between $1$ and $n$.
Let:
:$\mathbf B := \mathbf A \sqbrk {a_1, a_2, \ldots, a_k; b_1, b_2, \ldots, b_k}$
be the '''submatrix''' formed from rows $\set {a_1, a_2, \ldots, a_k}$ and columns $\set {b_1, b_2, \ldots, b_k}$.
The '''order-$k$ minor''' of $D$ formed from rows $r_1, r_2, \ldots, r_k$ and columns $s_1, s_2, \ldots, s_k$ can be denoted:
:$\map D {a_1, a_2, \ldots, a_k \mid b_1, b_2, \ldots, b_k}$.
Each element of $D$ is an order $1$ minor of $D$, and can be denoted:
:$\map D {a_i \mid b_j}$
\end{definition}