LU Decomposition and Scaling

I have a question about scaling and improving the solution of LU decomposition and final answer in Ax=b.

I am scaling the original matrix by an scale factor (e.g. 1024 or 1000), by doing this operation I am loosing some numbers which are close to denormal number and they are changing to 0.

So:

  1. By scaling it seems that I am destroing the original problem. How this process is helping?

  2. I have checked, sometimes the ||LU-PA|| is improving 1 digits but the ||Ax-b|| is not better or is worse. How it could be possible to have better LU decomposition but the acuuracy of ||Ax-b|| become worse?