A simple demonstration of the effects of an ill-conditioned matrix on the accuracy of obtaining the inverse of a matrix - and thus the subsequent solution of a system - can be found by considering the following matrix:
Obtain the subroutines dgeco.f, dgedi.f, dgesl.f, and dsvdc.f from the linpack directory of netlib. This can be accomplished by sending email to email@example.com with the line, send filename.f from linpack, in the text section. Netlib will automatically send the code back to you in an email message.
Note: If you have not used netlib before, you might want to look at the available software and read the section on netlib in the Linear Algebra chapter of CSEP.