Math 2.0

Linear and multilinear algebra; matrix theory Proposition

Determinant of a matrix with a zero row

Builds on Determinant

Statement

Let KK be a commutative ring and let AA be an n×nn \times n matrix over KK. If one row of AA is zero, then det⁡A=0\det A = 0, where det⁡\det is the determinant.

Remarks

This follows from linearity in that row: the zero row is 00 times itself, so det⁡A=0⋅det⁡A=0\det A = 0 \cdot \det A = 0. The same holds for a zero column.

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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 levelDeterminantdefinitionDeterminant of a matrix with a zero row, statement, no evidence levelDeterminant of a matr…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. Determinant of a matrix with a zero row statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Determinant, Basic rules
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