Definition:Cofactor

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 $n$.

Cofactor of an Element

Let $a_{rs}$ be an element of $D$.

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

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

$A_{rs} := \left({-1}\right)^{r+s} D_{rs}$

Cofactor of a Minor

Let $D \left({r_1, r_2, \ldots, r_k | s_1, s_2, \ldots, s_k}\right)$ be a order-$k$ minor of $D$.

Then the cofactor of $D \left({r_1, r_2, \ldots, r_k | s_1, s_2, \ldots, s_k}\right)$ can be denoted $\tilde D \left({r_1, r_2, \ldots, r_k | s_1, s_2, \ldots, s_k}\right)$ and is defined as:

$\tilde D \left({r_1, r_2, \ldots, r_k | s_1, s_2, \ldots, s_k}\right) = \left({-1}\right)^t D \left({r_{k+1}, r_{k+2}, \ldots, r_n | s_{k+1}, s_{k+2}, \ldots, s_n}\right)$

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, and 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).

Examples

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:

$D_{21} = \begin{vmatrix} a_{12} & a_{13} \\ a_{32} & a_{33}\end{vmatrix} = a_{12} a_{33} - a_{13} a_{32}$

(see that row 2 and column 1 have been deleted).

Thus:

$A_{21} = \left({-1}\right)^{3} \left({a_{12} a_{33} - a_{13} a_{32}}\right) = a_{13} a_{32} - a_{12} a_{33}$

Let:

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

Let $D \left({2, 3 | 2, 4}\right)$ be a order-$k$ minor of $D$.

Then:

$D \left({2, 3 | 2, 4}\right) = \begin{vmatrix} a_{22} & a_{24} \\ a_{32} & a_{34} \\ \end{vmatrix}$

and:

$ \tilde D \left({2, 3 | 2, 4}\right) = \left({-1}\right)^{2 + 3 + 2 + 4} D \left({1, 4 | 1, 3}\right) = - \begin{vmatrix} a_{11} & a_{13} \\ a_{41} & a_{43} \\ \end{vmatrix}$

Sources

• Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (1968): $\S 1.2.3$