Definition:Cofactor/Minor

Definition

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 $\map D {r_1, r_2, \ldots, r_k \mid s_1, s_2, \ldots, s_k}$ be a order-$k$ minor of $D$.

Then the cofactor of $\map D {r_1, r_2, \ldots, r_k \mid s_1, s_2, \ldots, s_k}$ can be denoted:

$\map {\tilde D} {r_1, r_2, \ldots, r_k \mid s_1, s_2, \ldots, s_k}$

and is defined as:

$\map {\tilde D} {r_1, r_2, \ldots, r_k \mid s_1, s_2, \ldots, s_k} = \paren {-1}^t \map D {r_{k + 1}, r_{k + 2}, \ldots, r_n \mid s_{k + 1}, s_{k + 2}, \ldots, s_n}$

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 $\set {r_1, r_2, \ldots, r_k}$

$s_{k + 1}, s_{k + 2}, \ldots, s_n$ are the numbers in $1, 2, \ldots, n$ not in $\set {s_1, s_2, \ldots, s_k}$

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).

Also known as

A cofactor is also known as a signed minor.