Math 2.0

Linear and multilinear algebra; matrix theory Proof

Proof that adding a multiple of one row to another preserves the determinant

Builds on Determinant is multilinear in the rowsDeterminant of a matrix with two equal rows

Proof

  1. Let aka_k be row kk of AA. Then row ii of BB is ai+caja_i + c a_j, and the other rows of BB are those of AA.
  2. Let A′A' be the matrix obtained from AA by replacing row ii with aja_j.
  3. The matrices AA, A′A' and BB agree outside row ii. Their rows ii are aia_i, aja_j and 1⋅ai+c⋅aj1 \cdot a_i + c \cdot a_j.
  4. By multilinearity in the rows with s=1s = 1 and t=ct = c, we get det⁡B=det⁡A+cdet⁡A′\det B = \det A + c \det A'.
  5. Rows ii and jj of A′A' are both aja_j, and i≠ji \ne j. So det⁡A′=0\det A' = 0, by the rule for two equal rows.
  6. By steps 4 and 5, det⁡B=det⁡A\det B = \det A.

Idea

Split the new row into its old part and the added part. The added part gives a matrix with a repeated row, and that determinant is zero.

Reading path

To understand this page from the ground up, read these 14 pages in order. The foundations come first. The path takes the proofs Proof that a matrix with two equal rows has determinant zero, Proof that the determinant is multilinear in the rows.

Show the path as a graph
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 a matrix with two equal rows has determinant zero, proof, no evidence levelProof that a matrix w…proofProof that the determinant is multilinear in the rows, proof, no evidence levelProof that the determ…proofDeterminant of a matrix with two equal rows, statement, no evidence levelDeterminant of a matr…statementDeterminant is multilinear in the rows, statement, no evidence levelDeterminant is multil…statementProof that adding a multiple of one row to another preserves the determinant, proof, no evidence levelProof that adding a m…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 a matrix with two equal rows has determinant zero proof
  11. Proof that the determinant is multilinear in the rows proof
  12. Determinant of a matrix with two equal rows statement
  13. Determinant is multilinear in the rows statement
  14. Proof that adding a multiple of one row to another preserves the determinant proof

Where this leads

4 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Determinant, Basic rules
About this page
TagVCTVF3AZ1Q
URIm2://VCTVF3AZ1Q
Kindproof
WritingW0 stub
MSC15A15
ProvesAdding a multiple of one row to another preserves the determinant
UsesDeterminant is multilinear in the rows, Determinant of a matrix with two equal rows
Changed4 November 2026

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