Cofactor (mathematics)

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In mathematics, a cofactor is a component of a matrix computation of the matrix determinant.

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

(−1)i+jdet⁡Mi,j.

The adjugate matrix adj M is the square matrix whose (i,j) entry is the (j,i) cofactor. We have

M⋅adjM=(det⁡M)In=adjM⋅M,

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 invertible, then M is invertible and the matrix inverse is determined as

M−1=(det⁡M)−1adjM.

## Example[edit]

Consider the following example matrix,

M=(a1a2a3b1b2b3c1c2c3).

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

M11=|b2b3c2c3|M12=|b1b3c1c3|M13=|b1b2c1c2|M21=|a2a3c2c3|M22=|a1a3c1c3|

M23=|a1a2c1c2|M31=|a2a3b2b3|M32=|a1a3b1b3|M33=|a1a2b1b2|

The adjugate matrix of M is

adjM=A=(M11−M21M31−M12M22−M32M13−M23M33),

and the inverse matrix is

M−1=|M|−1A.

Indeed,

(MM−1)11=|M|−1(a1M11−a2M12+a3M13)=|M||M|=1(MM−1)21=|M|−1(b1M11−b2M12+b3M13)=|M|−1[b1(b2c3−b3c2)−b2(b1c3−b3c1)+b3(b1c2−b2c1)]=0,

and the other matrix elements of the product follow likewise.

## References[edit]

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