Math 2.0

Linear and multilinear algebra; matrix theory Proposition

Scaling a row scales the determinant

Builds on Determinant

Statement

Let KK be a commutative ring, let AA be an n×nn \times n matrix over KK, and let c∈Kc \in K. Let BB be the matrix that results when one row of AA is multiplied by cc. Then det⁡B=cdet⁡A\det B = c \det A, where det⁡\det is the determinant.

Remarks

This is a special case of linearity in each row. It describes the effect of the scaling row operation on the determinant. Scaling all nn rows gives det⁡(cA)=cndet⁡A\det(cA) = c^n \det A, see the determinant of a scalar multiple.

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To understand this page from the ground up, read these 9 pages in order. The foundations come first.

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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 levelDeterminantdefinitionScaling a row scales the determinant, statement, no evidence levelScaling a row scales …statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Identity matrix object
  2. Matrix definition
  3. Vector space definition
  4. Square matrix definition
  5. Linear map definition
  6. Multilinear map definition
  7. Alternating multilinear map definition
  8. Determinant definition
  9. Scaling a row scales the determinant statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Determinant, Basic rules
About this page
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UsesDeterminant

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