\begin{definition}[Definition:Cofactor/Minor]
Let:
: $D = \begin{vmatrix}  a_{11} & a_{12} & \cdots & a_{1n} \\
  a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
  a_{n1} & a_{n2} & \cdots & a_{nn}\end{vmatrix}$
be a determinant of order $n$.
Let $D \left({r_1, r_2, \ldots, r_k \mid s_1, s_2, \ldots, s_k}\right)$ be a order-$k$ minor of $D$.
Then the '''cofactor''' of $D \left({r_1, r_2, \ldots, r_k \mid s_1, s_2, \ldots, s_k}\right)$ can be denoted:
:$\tilde D \left({r_1, r_2, \ldots, r_k \mid s_1, s_2, \ldots, s_k}\right)$
and is defined as:
:$\tilde D \left({r_1, r_2, \ldots, r_k \mid s_1, s_2, \ldots, s_k}\right) = \left({-1}\right)^t D \left({r_{k+1}, r_{k+2}, \ldots, r_n \mid s_{k+1}, s_{k+2}, \ldots, s_n}\right)$
where:
: $t = r_1 + r_2 + \ldots + r_k + s_1 + s_2 + \ldots s_k$
: $r_{k+1}, r_{k+2}, \ldots, r_n$ are the numbers in $1, 2, \ldots, n$ not in $\left\{{r_1, r_2, \ldots, r_k}\right\}$
: $s_{k+1}, s_{k+2}, \ldots, s_n$ are the numbers in $1, 2, \ldots, n$ not in $\left\{{s_1, s_2, \ldots, s_k}\right\}$
That is, the '''cofactor of a minor''' is the determinant formed from the rows and columns not in that minor, multiplied by the appropriate sign.
When $k = 1$, this reduces to the cofactor of an element (as above).
When $k = n$, the "minor" is in fact the whole determinant.
For convenience its '''cofactor''' is defined as being $1$.
Note that the '''cofactor''' of the '''cofactor''' of a minor is the minor itself (multiplied by the appropriate sign).
Category:Definitions/Determinants
\end{definition}