Definition:Minor of Determinant/Notation

Definition
Let $\mathbf A = \left[{a}\right]_n$ be a square matrix of order $n$.

Let $D := \det \left({\mathbf A}\right)$ denote the determinant of $\mathbf A$.

Let:
 * $\left\{ {a_1, a_2, \ldots, a_k}\right\}$ be the indices of the $k$ selected rows of $\mathbf A$
 * $\left\{ {b_1, b_2, \ldots, b_k}\right\}$ 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:
 * $\mathbf B := \mathbf A \left[{a_1, a_2, \ldots, a_k; b_1, b_2, \ldots, b_k}\right]$

be the submatrix formed from rows $\left\{ {a_1, a_2, \ldots, a_k}\right\}$ and columns $\left\{ {b_1, b_2, \ldots, b_k}\right\}$

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:
 * $D \left({a_1, a_2, \ldots, a_k \mid b_1, b_2, \ldots, b_k}\right)$.

Each element of $D$ is an order $1$ minor of $D$, and can be denoted:
 * $D \left({a_i \mid b_j}\right)$