Cofactor (linear algebra)

In linear algebra, the cofactor (sometimes called adjunct, see below) describes a particular construction that is useful for calculating both the determinant and inverse of square matrices. Specifically the cofactor of the (i, j) entry of a matrix, also known as the (i, j) cofactor of that matrix, is the signed minor of that entry.

## Informal approach to minors and cofactors

Finding the minors of a matrix A is a multi-step process:

1. Choose an entry aij from the matrix.
2. Cross out the entries that lie in the corresponding row i and column j.
3. Rewrite the matrix without the marked entries.
4. Obtain the determinant Mij of this new matrix.

Mij is termed the minor for entry aij.

If i + j is an even number, the cofactor Cij of aij coincides with its minor:

$C_{ij} = M_{ij}. \,$

Otherwise, it is equal to the additive inverse of its minor:

$C_{ij} = -M_{ij}. \,$

## Formal definition

If A is a square matrix, then the minor of its entry aij, also known as the i,j, or (i,j), or (i,j)th minor of A, is denoted by Mij and is defined to be the determinant of the submatrix obtained by removing from A its i-th row and j-th column.

It follows:

$C_{ij}=(-1)^{i+j} M_{ij} \,$

and Cij called the cofactor of aij, also referred to as the i,j, (i,j) or (i,j)th cofactor of A.

## Example

Given the matrix

$B = \begin{bmatrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \\ b_{31} & b_{32} & b_{33} \\ \end{bmatrix}$

suppose we wish to find the cofactor C23. The minor M23 is the determinant of the above matrix with row 2 and column 3 removed.

$M_{23} = \begin{vmatrix} b_{11} & b_{12} & \Box \\ \Box & \Box & \Box \\ b_{31} & b_{32} & \Box \\ \end{vmatrix}$ yields $M_{23} = \begin{vmatrix} b_{11} & b_{12} \\ b_{31} & b_{32} \\ \end{vmatrix} = b_{11}b_{32} - b_{31}b_{12}$

Using the given definition it follows that

$\ C_{23} = (-1)^{2+3}(M_{23})$
$\ C_{23} = (-1)^{5}(b_{11}b_{32} - b_{31}b_{12})$
$\ C_{23} = b_{31}b_{12} - b_{11}b_{32}.$

Note: the vertical lines are an equivalent notation for det(matrix)

## Cofactor expansion

Given the $n\times n$ matrix

$A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{bmatrix}$

The determinant of A (denoted det(A)) can be written as the sum of the cofactors of any row or column of the matrix multiplied by the entries that generated them.

Cofactor expansion along the jth column:

$\ \det(A) = a_{1j}C_{1j} + a_{2j}C_{2j} + a_{3j}C_{3j} + ... + a_{nj}C_{nj} = \sum_{i=1}^{n} a_{ij} C_{ij}$

Cofactor expansion along the ith row:

$\ \det(A) = a_{i1}C_{i1} + a_{i2}C_{i2} + a_{i3}C_{i3} + ... + a_{in}C_{in} = \sum_{j=1}^{n} a_{ij} C_{ij}$

## Matrix of cofactors

The matrix of cofactors for an $n\times n$ matrix A is the matrix whose (i,j) entry is the cofactor Cij of A. For instance, if A is

$A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{bmatrix}$

the cofactor matrix of A is

$C = \begin{bmatrix} C_{11} & C_{12} & \cdots & C_{1n} \\ C_{21} & C_{22} & \cdots & C_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ C_{n1} & C_{n2} & \cdots & C_{nn} \end{bmatrix}$

where Cij is the cofactor of aij.

The adjugate matrix is the transpose of the matrix of cofactors and is very useful due to its relation to the inverse of A.

$\mathbf{A}^{-1} = \frac{1}{\det \mathbf{A}} \mbox{adj}(\mathbf{A})$

The matrix of cofactors

$\begin{bmatrix} C_{11} & C_{12} & \cdots & C_{1n} \\ C_{21} & C_{22} & \cdots & C_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ C_{n1} & C_{n2} & \cdots & C_{nn} \end{bmatrix}$

when transposed becomes

$\mathrm{adj}(A) = \begin{bmatrix} C_{11} & C_{21} & \cdots & C_{n1} \\ C_{12} & C_{22} & \cdots & C_{n2} \\ \vdots & \vdots & \ddots & \vdots \\ C_{1n} & C_{2n} & \cdots & C_{nn} \end{bmatrix}.$

## A remark about different notations

In some books, including the so called "bible of matrix theory"[1] instead of cofactor the term adjunct is used. Moreover, it is denoted as Aij and defined in the same way as cofactor:

$\mathbf{A}_{ij} = (-1)^{i+j} \mathbf{M}_{ij}$

Using this notation the inverse matrix is written this way:

$\mathbf{A}^{-1} = \frac{1}{\det(A)}\begin{bmatrix} A_{11} & A_{21} & \cdots & A_{n1} \\ A_{12} & A_{22} & \cdots & A_{n2} \\ \vdots & \vdots & \ddots & \vdots \\ A_{1n} & A_{2n} & \cdots & A_{nn} \end{bmatrix}$

## References

1. ^ Felix Gantmacher, Theory of matrices (1st ed., original language is Russian), Moscow: State Publishing House of technical and theoretical literature, 1953, p.491,
• Anton, Howard; Rorres, Chris (2005), Elementary Linear Algebra (9th ed.), John Wiley and Sons, ISBN 0-471-66959-8

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Minor (linear algebra) — This article is about a concept in linear algebra. For the unrelated concept of minor in graph theory, see Minor (graph theory). In linear algebra, a minor of a matrix A is the determinant of some smaller square matrix, cut down from A by… …   Wikipedia

• Cofactor — may refer to any of the following: Cofactor (linear algebra), the signed minor of a matrix Minor (linear algebra), an alternative name for the determinant of a smaller matrix than that which it describes Cofactor (biochemistry), a substance that… …   Wikipedia

• List of mathematics articles (C) — NOTOC C C closed subgroup C minimal theory C normal subgroup C number C semiring C space C symmetry C* algebra C0 semigroup CA group Cabal (set theory) Cabibbo Kobayashi Maskawa matrix Cabinet projection Cable knot Cabri Geometry Cabtaxi number… …   Wikipedia

• Quadratic sieve — The quadratic sieve algorithm (QS) is a modern integer factorization algorithm and, in practice, the second fastest method known (after the general number field sieve). It is still the fastest for integers under 100 decimal digits or so, and is… …   Wikipedia

• Adjugate matrix — In linear algebra, the adjugate or classical adjoint of a square matrix is a matrix that plays a role similar to the inverse of a matrix; it can however be defined for any square matrix without the need to perform any divisions. The adjugate has… …   Wikipedia

• Cross product — This article is about the cross product of two vectors in three dimensional Euclidean space. For other uses, see Cross product (disambiguation). In mathematics, the cross product, vector product, or Gibbs vector product is a binary operation on… …   Wikipedia

• Determinant — This article is about determinants in mathematics. For determinants in epidemiology, see Risk factor. In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific… …   Wikipedia

• Curvilinear coordinates — Curvilinear, affine, and Cartesian coordinates in two dimensional space Curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian… …   Wikipedia

• Invertible matrix — In linear algebra an n by n (square) matrix A is called invertible (some authors use nonsingular or nondegenerate) if there exists an n by n matrix B such that where In denotes the n by n identity matrix and the multiplication used is ordinary… …   Wikipedia

• Laplace expansion — This article is about the expansion of the determinant of a square matrix as a weighted sum of determinants of sub matrices. For the expansion of an 1/r potential using spherical harmonical functions, see Laplace expansion (potential). In linear… …   Wikipedia