Math 2.0

Linear and multilinear algebra; matrix theory Definition

Cofactor

Builds on Minor

Definition

Let KK be a commutative ring and let AA be an n×nn \times n matrix over KK, with n≥2n \ge 2. The cofactor of AA in position (i,j)(i, j) is Cij=(−1)i+jMij,C_{ij} = (-1)^{i+j} M_{ij}, where MijM_{ij} is a first minor of AA. It is the minor that results when row ii and column jj are deleted.

Remarks

The signs (−1)i+j(-1)^{i+j} form a checkerboard pattern, with ++ in the top left corner. The cofactors give the Laplace expansion of the determinant. The transpose of the matrix of cofactors is the adjugate matrix.

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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 levelCofactordefinition
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  1. Identity matrix object
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  5. Linear map definition
  6. Multilinear map definition
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  9. Minor definition
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Further reading

  1. Web https://en.wikipedia.org/wiki/Minor_(linear_algebra), First minors
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