Math 2.0

Linear and multilinear algebra; matrix theory Proof

Proof that a matrix with a zero row has determinant zero

Builds on Scaling a row scales the determinant

Proof

  1. Let row ii of AA be zero.
  2. Multiplying row ii of AA by 00 gives a matrix whose row ii is 00 and whose other rows are those of AA. This matrix is AA itself.
  3. By the rule for scaling a row with c=0c = 0, we get det⁡A=0⋅det⁡A\det A = 0 \cdot \det A.
  4. So det⁡A=0\det A = 0.

Idea

A zero row is unchanged when it is multiplied by 00. Since the determinant is linear in that row, it must be multiplied by 00 too.

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 scaling a row scales the determinant, 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 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 …proofScaling a row scales the determinant, statement, no evidence levelScaling a row scales …statementProof that a matrix with a zero row has determinant zero, proof, no evidence levelProof that a matrix w…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
  13. Scaling a row scales the determinant statement
  14. Proof that a matrix with a zero row has determinant zero proof

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Determinant, Basic rules
About this page
Tag8T5B6HSHW7
URIm2://8T5B6HSHW7
Kindproof
WritingW0 stub
MSC15A15
ProvesDeterminant of a matrix with a zero row
UsesScaling a row scales the determinant
Changed4 November 2026

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