Mathematica incorporates the latest algorithms for solving industrial-scale linear systems, automatically switching between optimal dense and sparse algorithms—and handling exact, symbolic, and arbitrary-precision as well as machine-precision computation.
LinearSolve — solve a linear system, dense or sparse
LinearSolveFunction — a function created to repeatedly solve a linear system
Inverse — explicit dense matrix inverse
NullSpace — vectors spanning the null space of a matrix
RowReduce — generate a row echelon form
