proof of cofactor expansion

Let MmatN(K) be a n×n-matrix with entries from a commutativePlanetmathPlanetmathPlanetmathPlanetmath field K. Let e1,,en denote the vectors of the canonical basis of Kn. For the proof we need the following

Lemma: Let Mij* be the matrix generated by replacing the i-th row of M by ej. Then


where Mij is the (n-1)×(n-1)-matrix obtained from M by removing its i-th row and j-th column.


By adding appropriate of the i-th row of Mij* to its remaining rows we obtain a matrix with 1 at position (i,j) and 0 at positions (k,j) (ki). Now we apply the permutationMathworldPlanetmath


to rows and


to columns of the matrix. The matrix now looks like this:

  • Row/column 1 is the vector e1;

  • under row 1 and right of column 1 is the matrix Mij.

Since the determinantMathworldPlanetmath has changed its sign i+j-2 times, we have


Note also that only those permutations πSn are for the computation of the determinant of Mij* where π(i)=j. ∎

Now we start out with

detM =πSnsgnπ(j=1nmjπ(j))

From the previous lemma, it follows that the associated with Mik is the determinant of Mij*. So we have

Title proof of cofactor expansion
Canonical name ProofOfCofactorExpansion
Date of creation 2013-03-22 13:22:08
Last modified on 2013-03-22 13:22:08
Owner Thomas Heye (1234)
Last modified by Thomas Heye (1234)
Numerical id 13
Author Thomas Heye (1234)
Entry type Proof
Classification msc 15A15
Synonym Laplace expansionPlanetmathPlanetmath