Definition:Minor of Determinant

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

Let $$\det \left({\mathbf{A}}\right)$$ be the determinant of $$\mathbf{A}$$.

An order-$$k$$ minor of $$\det \left({\mathbf{A}}\right)$$ is a determinant of order $$k$$ obtained by deleting $$n-k$$ rows and $$n-k$$ columns from $$\det \left({\mathbf{A}}\right)$$.

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