Math 2.0

Linear and multilinear algebra; matrix theory Determinants Proposition

Determinant of a transpose

Builds on DeterminantTranspose

Statement

Let KK be a commutative ring and let AA be an n×nn \times n matrix over KK. Then det⁡AT=det⁡A\det A^T = \det A, where ATA^T is the transpose of AA and det⁡\det is the determinant.

Remarks

In the Leibniz formula, the term of σ\sigma for ATA^T equals the term of σ−1\sigma^{-1} for AA, and sgn⁡(σ−1)=sgn⁡(σ)\sgn(\sigma^{-1}) = \sgn(\sigma). So each property of the determinant for rows also holds for 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 matrixdefinitionTranspose, definition, no evidence levelTransposedefinitionLinear map, definition, no evidence levelLinear mapdefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionDeterminant, definition, no evidence levelDeterminantdefinitionDeterminant of a transpose, statement, no evidence levelDeterminant of a tran…statement
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  1. Identity matrix object
  2. Matrix definition
  3. Vector space definition
  4. Square matrix definition
  5. Transpose definition
  6. Linear map definition
  7. Multilinear map definition
  8. Alternating multilinear map definition
  9. Determinant definition
  10. Determinant of a transpose statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Determinant, Basic rules
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Tag0ZZQ2E1MTE
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Kindstatement, proposition
WritingW0 stub
MSC15A15
StatusThis copy of the site has no build data. Run m2 status 0ZZQ2E1MTE.
UsesDeterminant, Transpose

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