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LinearAlgebra[Modular][Determinant] - compute the determinant of a square mod m Matrix
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Calling Sequence
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Determinant(m, A, meth)
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Parameters
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m
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modulus
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A
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square mod m Matrix
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meth
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(optional) keyword for choice of method
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Description
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The Determinant function returns the mod m determinant of the input square mod m Matrix.
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The following methods are available:
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REF
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(default) Compute using standard row-reduction (Row Echelon Form)
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inplaceREF
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Compute using standard row-reduction in-place in the input Matrix
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RET
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Compute using a Row Echelon Transformation approach
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inplaceRET
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Compute using a Row Echelon Transformation in-place in the input Matrix
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Note that the two inplace methods available will destroy the data in the input Matrix, while the other methods will generate a copy of the Matrix in which to perform the computation.
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The RET methods are likely to be faster for large matrices, but may fail if the modulus is composite.
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This command is part of the LinearAlgebra[Modular] package, so it can be used in the form Determinant(..) only after executing the command with(LinearAlgebra[Modular]). However, it can always be used in the form LinearAlgebra[Modular][Determinant](..).
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Examples
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![with(LinearAlgebra[Modular])](/support/helpjp/helpview.aspx?si=4519/file06067/math123.png)
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![M := Create(p, 4, 4, random, float[8])](/support/helpjp/helpview.aspx?si=4519/file06067/math134.png)
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![M := Matrix([[96., 62., 93., 90.], [58., 16., 76., 19.], [7., 49., 42., 78.], [26., 19., 21., 21.]])](/support/helpjp/helpview.aspx?si=4519/file06067/math137.png)
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![5, Matrix([[96., 62., 93., 90.], [58., 16., 76., 19.], [7., 49., 42., 78.], [26., 19., 21., 21.]])](/support/helpjp/helpview.aspx?si=4519/file06067/math144.png)
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With inplaceREF the input Matrix is altered
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![5, Matrix([[1., 35., 4., 7.], [0., 1., 86., 38.], [0., 0., 1., 96.], [0., 0., 0., 1.]])](/support/helpjp/helpview.aspx?si=4519/file06067/math155.png)
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A composite example where the RET method is unsuccessful
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![M := Matrix([[3, 2], [2, 3]], datatype = integer)](/support/helpjp/helpview.aspx?si=4519/file06067/math161.png)
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![M := Matrix([[3, 2], [2, 3]])](/support/helpjp/helpview.aspx?si=4519/file06067/math164.png)
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| (6) |
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Note that this is only because this is a case where the row echelon form exists, but the row echelon transform cannot be written in the required form.
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