Definition:Minor of Determinant

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

Consider the order $k$ square submatrix $\mathbf B$ obtained by deleting $n - k$ rows and $n - k$ columns from $\mathbf A$.

Let $\det \left({\mathbf B}\right)$ denote the determinant of $\mathbf B$.

Then $\det \left({\mathbf B}\right)$ is an order-$k$ minor of $\det \left({\mathbf A}\right)$.

Thus a minor is a determinant formed from the elements (in the same relative order) of $k$ specified rows and columns.

Notation
Let $D$ be a determinant of order $n$.

An order-$k$ minor of $D$ whose elements are in rows $r_1, r_2, \ldots, r_k$ and columns $s_1, s_2, \ldots, s_k$ can be denoted $D \left({r_1, r_2, \ldots, r_k | s_1, s_2, \ldots, s_k}\right)$.

However, this is cumbersome for a minor of order $n-1$.

$D_{ij}$ denotes the minor of order $n-1$ obtained from $D$ by deleting all the elements of row $i$ and column $j$.

Each element of $D$ is an order 1 minor of $D$, and can (if you like) be denoted $D \left({i | j}\right)$.

Example
Let $D = \begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix}$.

Then $D \left({1, 2 | 1, 3}\right) = \begin{vmatrix} a_{11} & a_{13} \\ a_{21} & a_{23} \end{vmatrix}$.

Also see
The equivalent term in the context of a matrix is a submatrix.