Math 2.0

Linear and multilinear algebra; matrix theory Proposition

Determinant of a scalar multiple

Builds on DeterminantScalar multiplication of matrices

Statement

Let KK be a commutative ring, let AA be an n×nn \times n matrix over KK, and let c∈Kc \in K. Then det⁡(cA)=cndet⁡A,\det(cA) = c^n \det A, where cAcA is the scalar multiple and det⁡\det is the determinant.

Remarks

Each of the nn rows is multiplied by cc, and each row contributes one factor cc. For example, det⁡(−A)=(−1)ndet⁡A\det(-A) = (-1)^n \det A. So an n×nn \times n real skew-symmetric matrix with nn odd has determinant 00.

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Identity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionVector space, definition, no evidence levelVector spacedefinitionScalar multiplication of matrices, definition, no evidence levelScalar multiplication…definitionSquare 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 levelDeterminantdefinitionDeterminant of a scalar multiple, statement, no evidence levelDeterminant of a scal…statement
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  1. Identity matrix object
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  9. Determinant definition
  10. Determinant of a scalar multiple statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Determinant, Basic rules
About this page
Tag01EXSMK864
URIm2://01EXSMK864
Kindstatement, proposition
WritingW0 stub
MSC15A15
StatusThis copy of the site has no build data. Run m2 status 01EXSMK864.
UsesDeterminant, Scalar multiplication of matrices

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