## Infinitesimal Matrix Change

Let B, A, and e be square matrices with e small, and define

 (1)

where I is the Identity Matrix. Then the inverse of is approximately
 (2)

This can be seen by multiplying
 (3)

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

 (4)

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.

 (5)

so
 (6)

Using a general Matrix Inverse identity then gives
 (7)