determinant as a multilinear mapping

Let ๐Œ=(Miโขj) be an nร—n matrix with entries in a field K. The matrix ๐Œ is really the same thing as a list of n column vectorsMathworldPlanetmath of size n. Consequently, the determinantMathworldPlanetmath operationMathworldPlanetmath may be regarded as a mapping

det:Knร—โ€ฆร—Knโžnโขย timesโ†’K

The determinant of a matrix ๐Œ is then defined to be detโก(๐Œ1,โ€ฆ,๐Œn), where ๐ŒjโˆˆKn denotes the jth column of ๐Œ.

Starting with the definition

detโก(๐Œ1,โ€ฆ,๐Œn)=โˆ‘ฯ€โˆˆSnsgnโข(ฯ€)โขM1โขฯ€1โขM2โขฯ€2โขโ‹ฏโขMnโขฯ€n (1)

the following properties are easily established:

  1. 1.

    the determinant is multilinear;

  2. 2.

    the determinant is anti-symmetric;

  3. 3.

    the determinant of the identity matrixMathworldPlanetmath is 1.

These three properties uniquely characterize the determinant, and indeed can โ€” some would say should โ€” be used as the definition of the determinant operation.

Let us prove this. We proceed by representing elements of Kn as linear combinationsMathworldPlanetmath of


the standard basis of Kn. Let ๐Œ be an nร—n matrix. The jth column is represented as โˆ‘iMiโขjโข๐ži; whence using multilinearity

detโก(๐Œ) =detโก(โˆ‘iMiโข1โข๐ži,โˆ‘iMiโข2โข๐ži,โ€ฆ,โˆ‘iMiโขnโข๐ži)

The anti-symmetry assumptionPlanetmathPlanetmath implies that the expressions detโก(๐ži1,๐ži2,โ€ฆ,๐žin) vanish if any two of the indices i1,โ€ฆ,in coincide. If all n indices are distinct,


the sign in the above expression being determined by the number of transpositionsMathworldPlanetmath required to rearrange the list (i1,โ€ฆ,in) into the list (1,โ€ฆ,n). The sign is therefore the parity of the permutationMathworldPlanetmath (i1,โ€ฆ,in). Since we also assume that


we now recover the original definition (1).

Title determinant as a multilinear mapping
Canonical name DeterminantAsAMultilinearMapping
Date of creation 2013-03-22 13:09:17
Last modified on 2013-03-22 13:09:17
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 5
Author rmilson (146)
Entry type Theorem
Classification msc 15A15
Related topic ExteriorAlgebra