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) = $\displaystyle x+\Delta x + i (y+\Delta y)$

Where x and y are exact values for the real and imaginary part. And $\displaystyle \Delta x, \Delta y$ are absolute errors.

Than the relative errors should be:

$\displaystyle \delta_x := \frac{\Delta x}{x} \leq \epsilon \cdot cond(A)$

$\displaystyle \delta_y := \frac{\Delta y}{y} \leq \epsilon \cdot cond(A)$

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

$\displaystyle \delta_{y/x} \leq \delta_x + \delta_y$

In this case $\displaystyle \epsilon \cdot cond(A) = 3.8925e-04$

So: $\displaystyle \delta_{y/x} \leq 7.7849e-04 $

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.