Category:Examples of Cofactors

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This category contains examples of Cofactor.

Let $R$ be a commutative ring with unity.

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

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 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).

Pages in category "Examples of Cofactors"

The following 5 pages are in this category, out of 5 total.