Math 2.0

statement proposition

Determinant of a matrix with a zero row

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.

Explanations

L1 human

A zero row means that one edge of the box spanned by the rows has length zero. Such a box has no volume. So the determinant is zero.

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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
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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

Sources

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

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