# Cofactor (mathematics)

In mathematics, a **cofactor** appears in the definition of the determinant of a square matrix.

Let *M* be a square matrix of size *n*. The (*i*,*j*) **minor** refers to the determinant of the (*n*-1)×(*n*-1) submatrix *M*_{i,j} formed by deleting the *i*-th row and *j*-th column from *M* (or sometimes just to the submatrix *M*_{i,j} itself). The corresponding *cofactor* is the signed minor

The **adjugate matrix** adj *M* (in older literature called **adjoint matrix**^{[1]}) is the *n*×*n* matrix whose (*i*,*j*) entry is the (*j*,*i*) cofactor (note the transposition of the indices). Letting *I*_{n} be the *n*×*n* identity (unit) matrix, we have

which encodes the rule for expansion of the determinant of *M* by any the cofactors of any row or column.
This expression shows that if det(*M*) is non-zero, then *M* is invertible and its inverse is the following,

A proof of this equation may be found in this article.

## [edit] Example

Consider the following example matrix,

Its minors are the determinants (vertical bars indicate a determinant):

The adjugate matrix of *M* is

and the inverse matrix is

Indeed,

and the other matrix elements of the product follow likewise.

## [edit] Note

- ↑ The term "adjoint" for the adjugate matrix is disappearing because it is felt that it is easily confused with Hermitian adjoint, the transpose and complex conjugate of a matrix.

## [edit] References

- C.W. Norman (1986).
*Undergraduate Algebra: A first course*. Oxford University Press, 306,310,315. ISBN 0-19-853248-2.

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