Math 2.0

Linear and multilinear algebra; matrix theory Determinants Theorem

Laplace expansion along a row

Builds on DeterminantCofactor

Statement

Let KK be a commutative ring, let n≥2n \ge 2, and let A=(aij)A = (a_{ij}) be an n×nn \times n matrix over KK. Let CijC_{ij} be the cofactors of AA. Then, for each row ii, det⁡A=∑j=1naijCij=∑j=1n(−1)i+jaijMij.\det A = \sum_{j=1}^{n} a_{ij} C_{ij} = \sum_{j=1}^{n} (-1)^{i+j} a_{ij} M_{ij}.

Remarks

The expansion reduces a determinant of order nn to nn determinants of order n−1n - 1. It is efficient when the chosen row contains many zeros. There is a similar expansion along a column. Replace the entries of row ii by those of another row kk. Then the sum is 00, because the new matrix has two equal rows.

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Identity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionVector space, definition, no evidence levelVector spacedefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionLinear map, definition, no evidence levelLinear mapdefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionDeterminant, definition, no evidence levelDeterminantdefinitionMinor, definition, no evidence levelMinordefinitionCofactor, definition, no evidence levelCofactordefinitionLaplace expansion along a row, statement, no evidence levelLaplace expansion alo…statement
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  1. Identity matrix object
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  9. Minor definition
  10. Cofactor definition
  11. Laplace expansion along a row statement

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Further reading

  1. Web https://en.wikipedia.org/wiki/Laplace_expansion, lead section
About this page
Tag0C08BQ3M05
URIm2://0C08BQ3M05
Aliasescofactor-expansion
Kindstatement, theorem
WritingW0 stub
MSC15A15
StatusThis copy of the site has no build data. Run m2 status 0C08BQ3M05.
UsesDeterminant, Cofactor

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