LinearSolve[m, b, ZeroTest->test] evaluates test[m[[i, j]]] to determine whether matrix elements are zero. The default setting is ZeroTest->Automatic.
A Method option can also be given. Settings for exact and symbolic matrices include "CofactorExpansion", "DivisionFreeRowReduction" and "OneStepRowReduction". Settings for approximate numerical matrices include "Cholesky", and for sparse arrays "Multifrontal" and "Krylov". The default setting of Automatic switches among these methods depending on the matrix given.