 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.
Contents
Informal approach to minors and cofactors
Finding the minors of a matrix A is a multistep process:
 Choose an entry a_{ij} from the matrix.
 Cross out the entries that lie in the corresponding row i and column j.
 Rewrite the matrix without the marked entries.
 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:
Otherwise, it is equal to the additive inverse of its minor:
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 ith row and jth column.
It follows:
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
suppose we wish to find the cofactor C_{23}. The minor M_{23} is the determinant of the above matrix with row 2 and column 3 removed.
 yields
Using the given definition it follows that
Note: the vertical lines are an equivalent notation for det(matrix)
Cofactor expansion
Main article: Laplace expansionGiven the matrix
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:
Cofactor expansion along the ith row:
Matrix of cofactors
The matrix of cofactors for an matrix A is the matrix whose (i,j) entry is the cofactor C_{ij} of A. For instance, if A is
the cofactor matrix of A is
where C_{ij} is the cofactor of a_{ij}.
Adjugate
Main article: Adjugate matrixThe adjugate matrix is the transpose of the matrix of cofactors and is very useful due to its relation to the inverse of A.
The matrix of cofactors
when transposed becomes
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 A_{ij} and defined in the same way as cofactor:
Using this notation the inverse matrix is written this way:
Keep in mind that adjunct is not adjugate or adjoint.
See also
References
 ^ 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 0471669598
External links
 MIT Linear Algebra Lecture on Cofactors at Google Video, from MIT OpenCourseWare
 PlanetMath
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