Math 2.0

Linear and multilinear algebra; matrix theory Definition

Determinant

Builds on Alternating multilinear mapSquare matrixIdentity matrix

Definition

Let KK be a commutative ring and let n≥1n \ge 1. There is exactly one map det⁡ ⁣:Mn(K)→K\det \colon M_n(K) \to K with these properties:

  1. det⁡A\det A is a multilinear function of the nn rows of AA.
  2. det⁡A=0\det A = 0 when two rows of AA are equal.
  3. det⁡In=1\det I_n = 1, for the identity matrix InI_n.

This map is the determinant.

Remarks

In other words, det⁡\det is the unique alternating multilinear map of the rows with value 11 on InI_n. The Leibniz formula gives det⁡A\det A as a sum over permutations. For n=2n = 2, det⁡(abcd)=ad−bc\det \begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc. The same characterization holds with columns in place of rows, because det⁡AT=det⁡A\det A^T = \det A.

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Further reading

  1. Web https://en.wikipedia.org/wiki/Determinant, As an alternating multilinear form
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Kinddefinition
WritingW1 stated
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DefinesDeterminant
UsesAlternating multilinear map, Square matrix, Identity matrix

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