Important: The linalg package has been deprecated. Use the superseding command LinearAlgebra[GaussianElimination], instead.

- For information on migrating linalg code to the new packages, see examples/LinearAlgebraMigration.

Gaussian elimination with row pivoting is performed on A, an n by m matrix over a field. At present, if the matrix contains floating-point or decimal numbers, then Gaussian elimination with partial pivoting is used where all arithmetic is done in floating-point at Digits precision.  In this case, the matrix entries on input must all be numbers of type numeric or complex(numeric). Otherwise ordinary Gaussian elimination is used.  At present, the matrix entries must be rationals or complex rationals or in general rational functions with these coefficients.

The result is an upper triangular matrix B. If A is an n by n matrix then $\prod _{i\],i=1\mathrm{..}n}B\[i$.

If an optional second parameter is specified, and it is a name, it is assigned the rank of A. The rank of A is the number of non-zero rows in the resulting matrix.

If an optional third parameter is also specified, and the rank of A = n, then it is assigned the determinant of $\mathrm{submatrix}\left(A\,1..n\,1..n\right)$.

If an optional second parameter is specified, and it is an integer, the elimination is terminated at this column position.

The command with(linalg,gausselim) allows the use of the abbreviated form of this command.

$\mathrm{with}\left(\mathrm{linalg}\right)\:$

$A≔\mathrm{matrix}\left(3\,3\,\left[x\,1\,0\,0\,0\,1\,1\,y\,1\right]\right)$

$A≔\left[\begin{array}{ccc}x& 1& 0\\ 0& 0& 1\\ 1& y& 1\end{array}\right]$

$\mathrm{gausselim}\left(A\,'r'\,'d'\right)$

$\left[\begin{array}{ccc}x& 1& 0\\ 0& \frac{yx-1}{x}& 1\\ 0& 0& 1\end{array}\right]$

$\mathrm{rank}\left(A\right)$

$3$

$\det \left(A\right)$

$-yx+1$