Needs["LinearAlgebra`"]
SmallA  = N[ HilbertMatrix[5]];
Smallb = 
        Table[
Random[], 
{5}];
Export["SmallA.mat", SmallA]
Export["Smallb.mat", Smallb]

Needs["LinearAlgebra`"]
BigA  = 
        Table[ Random[], {100}, {100}];
Bigb = 
        Table[
Random[], 
{100}];
Export["BigA.mat", BigA]
Export["Bigb.mat", Bigb]

BigA = Import["BigA.mat"];
Bigb = 
      Import["Bigb.mat"]; LinearSolve[ BigA, 
        Bigb][[23]]