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 $a_{ij}$ 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 $M_{ij}$ of this new matrix.

$M_{ij}$ is termed the minor for entry $a_{ij}$.

If i + j is an even number, the cofactor $C_{ij}$ of $a_{ij}$ 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 $a_{ij}$, also known as the i,j, or (i,j), or (i,j)^th minor of A, is denoted by $M_{ij}$ 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 $C_{ij}$ called the cofactor of $a_{ij}$, 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

Main article: Laplace 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 C_ij 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 C_ij is the cofactor of a_ij.

Adjugate

Main article: Adjugate matrix

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" instead of cofactor the term adjunct is used. Moreover, it is denoted as A_ij 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:

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

Keep in mind that adjunct is not adjugate or adjoint.

See also

• Linear algebra
• Matrix (mathematics)
• Minor (linear algebra)
• Adjugate matrix

References

• Anton, Howard; Rorres, Chris (2005), Elementary Linear Algebra (9th ed.), John Wiley and Sons, ISBN 0-471-66959-8 

External links

• MIT Linear Algebra Lecture on Cofactors at Google Video, from MIT OpenCourseWare
• PlanetMath