Sparse matrix computational difficulties

Question

I am a computer science student. But I start writing here because my problem is not code related. It is purely computational related.

I am trying to calculated inverse of a large (e.g., $2000 \times 2000$) square matrix. This matrix is band diagonal matrix.

The non zero elements are very small numbers (e.g: $1--6$). They never exceed 6.

Now when I invert this matrix the result of inversion are all same number displayed.

Output:

9.711E013  9.711E013  9.711E013  9.711E013....................
9.711E013  9.711E013  9.711E013  9.711E013......................

9.711E013  9.711E013  9.711E013  9.711E013.....................
..............................................................

 9.711E013  9.711E013  9.711E013  9.711E013

My question is that why this type of output I get? What is the mathematical reason behind this?

Answer 1

EDITED: thanks to comment of copper.hat.

The mathematical/numerical reason is that the condition number defined as $$ \varkappa(A) = \frac{\sigma_{\max}(A)}{\sigma_{\min}(A)} $$ is too large, say of order $10^{10}$ or even more. $\sigma_{\min, \max}(A)$ is the minimal and maximal singular values of matrix $A$.

The case of small determinant is a particular case and in some cases implied from the argument above.