Math 2.0

Linear and multilinear algebra; matrix theory Proof

Proof that scaling a row scales the determinant

Builds on Determinant is multilinear in the rows

Proof

  1. Let ii be the row of AA that is multiplied by cc, and let aa be row ii of AA. Then row ii of BB is caca.
  2. The matrices AA, AA and BB agree outside row ii. Their rows ii are aa, aa and ca=c⋅a+0⋅aca = c \cdot a + 0 \cdot a.
  3. By multilinearity in the rows with s=cs = c and t=0t = 0, we get det⁡B=cdet⁡A+0⋅det⁡A\det B = c \det A + 0 \cdot \det A.
  4. So det⁡B=cdet⁡A\det B = c \det A.

Idea

Linearity in one row includes homogeneity: a scalar factor in that row comes out of the determinant. We get it from the linearity rule by taking the second coefficient to be zero.

Reading path

To understand this page from the ground up, read these 12 pages in order. The foundations come first. The path takes the proof Proof that the determinant is multilinear in the rows.

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Identity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionMultilinear map, concept, no evidence levelMultilinear mapconceptVector 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 levelDeterminantdefinitionProof that the determinant is multilinear in the rows, proof, no evidence levelProof that the determ…proofDeterminant is multilinear in the rows, statement, no evidence levelDeterminant is multil…statementProof that scaling a row scales the determinant, proof, no evidence levelProof that scaling a …proof
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.
  • no evidence level
  1. Identity matrix object
  2. Matrix definition
  3. Multilinear map concept
  4. Vector space definition
  5. Square matrix definition
  6. Linear map definition
  7. Multilinear map definition
  8. Alternating multilinear map definition
  9. Determinant definition
  10. Proof that the determinant is multilinear in the rows proof
  11. Determinant is multilinear in the rows statement
  12. Proof that scaling a row scales the determinant proof

Where this leads

14 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Determinant, Basic rules
About this page
Tag90DTG4Q0EA
URIm2://90DTG4Q0EA
Kindproof
WritingW0 stub
MSC15A15
ProvesScaling a row scales the determinant
UsesDeterminant is multilinear in the rows
Changed4 November 2026

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