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.

Fraction-free Gaussian elimination with row pivoting is performed on A, an n by m matrix of multivariate polynomials over the rationals. The result is an upper triangular matrix of multivariate polynomials.

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,ffgausselim) allows the use of the abbreviated form of this command.

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

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

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

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

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

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

$3$

$\det \left(A\right)$

$-yx+1$