cofactor expansion

Let M be an n×n matrix with entries Mij that are elements of a commutative ring. Let mij denote the determinantDlmfMathworldPlanetmath of the (n-1)×(n-1) submatrixMathworldPlanetmath obtained by deleting row i and column j of M, and let


The subdeterminants mij are called the minors of M, and the Cij are called the cofactors.

We have the following useful formulas for the cofactors of a matrix. First, if we regard detM as a polynomial in the entries Mij, then we may write

Cij=MMij (1)

Second, we may regard the determinant of M=(M1,,Mn) as a multi-linear, skew-symmetric functionMathworldPlanetmath of its columns:


This point of view leads to the following formula:

Cij=det(M1,,Mj^,𝐞i,,Mn), (2)

where the notation indicates that column j has been replaced by the ith standard vector.

As a consequence, we obtain the following representation of the determinant in terms of cofactors:

det(M) =det(M1,,M1j𝐞1++Mnj𝐞n,,Mn)

The above identityPlanetmathPlanetmath is often called the cofactor expansion of the determinant along column j. If we regard the determinant as a multi-linear, skew-symmetric function of n row-vectors, then we obtain the analogous cofactor expansion along a row:

det(M) =i=1nMjiCji.


Consider a general 3×3 determinant


The above can equally well be expressed as a cofactor expansion along the first row:

|a1a2a3b1b2b3c1c2c3| =a1|b2b3c2c3|-a2|b1b3c1c3|+a3|b1b2c1c2|

or along the second column:

|a1a2a3b1b2b3c1c2c3| =-a2|b1b3c1c3|+b2|a1a3c1c3|-c2|a1a3b1b3|

or indeed as four other such expansion corresponding to rows 2 and 3, and columns 1 and 3.

Title cofactor expansion
Canonical name CofactorExpansion
Date of creation 2013-03-22 12:01:07
Last modified on 2013-03-22 12:01:07
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 17
Author rmilson (146)
Entry type Theorem
Classification msc 15A15
Synonym Laplace expansion
Synonym cofactor
Synonym minor
Synonym subdeterminant
Related topic SarrusRule