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Gausselim

inert Gaussian elimination

Gaussjord

inert Gauss Jordan elimination

 Calling Sequence Gausselim(A) mod p Gaussjord(A) mod p Gausselim(A, 'r', 'd') mod p Gaussjord(A, 'r', 'd') mod p

Parameters

 A - Matrix 'r' - (optional) for returning the rank of A 'd' - (optional) for returning the determinant of A 'p' - an integer, the modulus

Description

 • The Gausselim and Gaussjord functions are placeholders for representing row echelon forms of the rectangular matrix A.
 • The commands Gausselim(A,...) mod p and Gassjord(A,...) mod p apply Gaussian elimination with row pivoting to A, a rectangular matrix over a finite ring of characteristic p. This includes finite fields, GF(p), the integers mod p, and GF(p^k) where elements of GF(p^k) are expressed as polynomials in RootOfs.
 • The result of the Gausselim command is a an upper triangular matrix B in row echelon form.  The result of the Gaussjord command is also an upper triangular matrix B but in reduced row echelon form.
 • If an optional second parameter is specified, and it is a name, it is assigned the rank of the matrix A.
 • If A is an $m$ by $n$ matrix with $m\le n$ and if an optional third parameter is also specified, and it is a name, it is assigned the determinant of the matrix A[1..m,1..m].

Examples

 > A := Matrix([[1,2,3],[1,3,0],[1,4,3]]);
 ${A}{≔}\left[\begin{array}{ccc}{1}& {2}& {3}\\ {1}& {3}& {0}\\ {1}& {4}& {3}\end{array}\right]$ (1)
 > Gausselim(A) mod 5;
 $\left[\begin{array}{ccc}{1}& {2}& {3}\\ {0}& {1}& {2}\\ {0}& {0}& {1}\end{array}\right]$ (2)
 > B := ArrayTools[Concatenate](2,A,LinearAlgebra[IdentityMatrix](3));
 ${B}{≔}\left[\begin{array}{cccccc}{1}& {2}& {3}& {1}& {0}& {0}\\ {1}& {3}& {0}& {0}& {1}& {0}\\ {1}& {4}& {3}& {0}& {0}& {1}\end{array}\right]$ (3)
 > Gaussjord(B) mod 5;
 $\left[\begin{array}{cccccc}{1}& {0}& {0}& {4}& {1}& {1}\\ {0}& {1}& {0}& {2}& {0}& {3}\\ {0}& {0}& {1}& {1}& {3}& {1}\end{array}\right]$ (4)
 > Inverse(A) mod 5;
 $\left[\begin{array}{ccc}{4}& {1}& {1}\\ {2}& {0}& {3}\\ {1}& {3}& {1}\end{array}\right]$ (5)
 > alias(a=RootOf(x^4+x+1) mod 2): # GF(2^4)
 > A := Matrix([[1,a,a^2],[a,a^2,a^3],[a^2,a^3,1]]);
 ${A}{≔}\left[\begin{array}{ccc}{1}& {a}& {{a}}^{{2}}\\ {a}& {{a}}^{{2}}& {{a}}^{{3}}\\ {{a}}^{{2}}& {{a}}^{{3}}& {1}\end{array}\right]$ (6)
 > Gausselim(A,'r','d') mod 2;
 $\left[\begin{array}{ccc}{1}& {a}& {{a}}^{{2}}\\ {0}& {0}& {a}\\ {0}& {0}& {0}\end{array}\right]$ (7)
 > r;
 ${2}$ (8)
 > d;
 ${0}$ (9)