Math 2.0

statement proposition

Determinant of a matrix with two equal rows

Statement

Let KK be a commutative ring and let AA be an n×nn \times n matrix over KK. If two different rows of AA are equal, then det⁡A=0\det A = 0, where det⁡\det is the determinant.

Remarks

This is property 2 of the definition of the determinant. The same holds for two equal columns. Over a field, more generally, det⁡A=0\det A = 0 when the rows of AA are linearly dependent.

Explanations

L1 human

Two equal rows mean that the rows span less than the full space. Then the box spanned by the rows is flat, and its volume is zero. 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 two equal rows, statement, no evidence levelDeterminant of a matr…statement
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  1. Identity matrix object
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  9. Determinant of a matrix with two equal rows statement

Sources

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

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