Random Sparse Matrices

n=1230;
A=SparseArray[{
Band[{1,1}]:>RandomReal[{3,12}]
},{n,n}];
nz=5000;
Do[
{i,j}=RandomInteger[{1,n},2];
A[[i,j]]+=RandomReal[{-1,1}],
{nz}];
{LU,P,cond}=LUDecomposition[A];
P==Range[1,n]
Map[MatrixPlot[#,PlotLabel->Length[ArrayRules[#]]]&,
 {A,LU}]