Math 2.0

Linear and multilinear algebra; matrix theory Proposition

Permuting rows multiplies the determinant by the sign

Builds on Determinant

Statement

Let KK be a commutative ring, let A=(aij)A = (a_{ij}) be an n×nn \times n matrix over KK, and let σ\sigma be a permutation of {1,…,n}\{1, \dots, n\}. Let B=(bij)B = (b_{ij}) be the matrix with bij=aσ(i),jb_{ij} = a_{\sigma(i), j}. Then det⁡B=sgn⁡(σ)det⁡A\det B = \sgn(\sigma) \det A, where det⁡\det is the determinant.

Remarks

In particular, exchanging two rows changes the sign of the determinant. For a permutation matrix PP, this gives det⁡P=±1\det P = \pm 1. The same holds for a permutation of the columns.

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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 levelDeterminantdefinitionPermuting rows multiplies the determinant by the sign, statement, no evidence levelPermuting rows multip…statement
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  1. Identity matrix object
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  6. Multilinear map definition
  7. Alternating multilinear map definition
  8. Determinant definition
  9. Permuting rows multiplies the determinant by the sign statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Determinant, Basic rules
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