Infinitesimal Matrix Change

Let , , and be square matrices with small, and define

where is the identity matrix. Then the inverse of is approximately

This can be seen by multiplying

Note that if we instead let , and look for an inverse of the form , we obtain

In order to eliminate the term, we require . However, then , so so there can be no inverse of this form.

The exact inverse of can be found as follows.

so

Using a general matrix inverse identity then gives