Infinitesimal Matrix Change

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

(1)

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

(2)

This can be seen by multiplying

			(3)
			(4)
			(5)
			(6)

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

			(7)
			(8)
			(9)
			(10)

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.

(11)

so

(12)

Using a general matrix inverse identity then gives

(13)