I'm having numerical problem after inverting complex matrix.

I have sparse, complex matrix A{f}, invert it using the following command (MATLAB):

Where I denote identity matrix and f denotes some parameter (not important in this case).Code:Ainv = A{f}\I;

Than I focus on the particular element of the inverse matrix. Let's say Ainv(1, 1). I observed that for some values of parameter f the results are wrong (diffrent than I expected).

Ainv(1, 1) has a good value, butif I take imag(Ainv(1, 1))/real(Ainv(1, 1)) i get rubish!

I belive the causes of this strange behaviour could be:

1. MATLAB algorithm is unstable for this matrix

2. The matrix is ill-conditioned, but the problem occours only if I take ratio of the real and imaginary part.

I focused on the 2. because there is nothing to do about Matlab algorithm.

The condition number of the matrix is rather large:

cond(A) = 1.7530e+12

Using this number I'm trying to estimate relative error.And here comes the problem, because I don't really know how to do that.

If I don't misunderstand the concept of condition number, it should be done like that.

Let Ainv(1, 1) =

Where x and y are exact values for the real and imaginary part. And are absolute errors.

Than the relative errors should be:

As far as I know the relative errors are added to each other when I'm doing division. So I should get:

In this case

So:

Thus the relative error should be far less than 1%, (and I get far more error than 100%). So what causes the problems than??

Maybe I don't understand the conception of condition number. If so, correct me and tell me how to estimate error in this case.

Thanks in advance.