Math 2.0

Linear and multilinear algebra; matrix theory Determinants Proposition

Determinant of a triangular matrix

Builds on DeterminantUpper triangular matrix

Statement

Let KK be a commutative ring and let A=(aij)A = (a_{ij}) be an n×nn \times n upper or lower triangular matrix over KK. Then det⁡A=a11a22⋯ann,\det A = a_{11} a_{22} \cdots a_{nn}, where det⁡\det is the determinant.

Remarks

In the Leibniz formula, all terms except the term of the identity permutation contain an entry below, or above, the diagonal, so they are zero. Row reduction to a triangular matrix, with this rule, is a practical way to compute determinants. A triangular matrix over a field is invertible if and only if all its diagonal entries are nonzero.

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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 mapdefinitionUpper triangular matrix, definition, no evidence levelUpper triangular matr…definitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionDeterminant, definition, no evidence levelDeterminantdefinitionDeterminant of a triangular matrix, statement, no evidence levelDeterminant of a tria…statement
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  10. Determinant of a triangular matrix statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Triangular_matrix, Properties
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