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:


 * $$A_{rs} \ \stackrel {\mathbf {def}} {=\!=} \ \left({-1}\right)^{r+s} D_{rs}$$

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