Definition:Cofactor

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Definition

Let $R$ be a commutative ring with unity.

Let $\mathbf A \in R^{n \times n}$ be a square matrix of order $n$.

Let:

$\quad D = \begin{vmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn}\end{vmatrix}$ be the determinant of $\mathbf A$.


Cofactor of an Element

Let $a_{r s}$ be an element of $D$.

Let $D_{r s}$ be the determinant of order $n-1$ obtained from $D$ by deleting row $r$ and column $s$.


Then the cofactor $A_{r s}$ of the element $a_{r s}$ is defined as:

$A_{r s} := \paren {-1}^{r + s} D_{r s}$


Cofactor of a Minor

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.


Examples

Arbitrary Example 1

Let $D$ be the determinant defined as:

$D = \begin {vmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end {vmatrix}$


Then the cofactor of $2$ is defined as:

\(\ds D_{12}\) \(=\) \(\ds \begin {vmatrix} 4 & 6 \\ 7 & 9 \end {vmatrix}\)
\(\ds \) \(=\) \(\ds \paren {-1}^{2 + 1} \paren {4 \times 9 - 6 \times 7}\)
\(\ds \) \(=\) \(\ds 6\)


Arbitrary Example 2

Let $D$ be the determinant defined as:

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


Then the cofactor of $a_{2 1}$ is defined as:

\(\ds A_{21}\) \(=\) \(\ds \paren {-1}^3 D_{21}\)
\(\ds \) \(=\) \(\ds \paren {-1}^3 \begin {vmatrix} a_{12} & a_{13} \\ a_{32} & a_{33} \end {vmatrix}\)
\(\ds \) \(=\) \(\ds -\paren {a_{12} a_{33} - a_{13} a_{32} }\)
\(\ds \) \(=\) \(\ds a_{13} a_{32} - a_{12} a_{33}\)


Arbitrary Example 3

Let $D$ be the determinant defined as:

$\quad D = \begin{vmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \\ \end{vmatrix}$


Let $\map D {2, 3 \mid 2, 4}$ be an order-$2$ minor of $D$.

Then the cofactor of $\map D {2, 3 \mid 2, 4}$ is given by:

\(\ds \map {\tilde D} {2, 3 \mid 2, 4}\) \(=\) \(\ds \paren {-1}^{2 + 3 + 2 + 4} \map D {1, 4 \mid 1, 3}\)
\(\ds \) \(=\) \(\ds \paren {-1}^{11} \begin {vmatrix} a_{11} & a_{13} \\ a_{41} & a_{43} \\ \end {vmatrix}\)
\(\ds \) \(=\) \(\ds -\paren {a_{11} a_{43} - a_{41} a_{13} }\)
\(\ds \) \(=\) \(\ds a_{41} a_{13} - a_{11} a_{43}\)


Also see

  • Results about cofactors can be found here.


Sources

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