Computing the derivative of an inverse matrix

[Demon117]

Homework Statement

If A, B are elements of Mat(n, R) and A is invertible, compute

$$\frac{d}{dt}_{t=0}(A+tB)^{-1}$$

The Attempt at a Solution

The derivative will be of the form

$$\frac{d}{dt}(A+tB)^{-1}=-(A+tB)^{-1}\frac{d}{dt}((A+tB))(A+tB)^{-1}$$

but I need to evaluate this at t=0, so how do I simplify the expression on the right hand side? Since d/dt is a linear operator do I just attack each term individually, that is, take a derivative of A with respect to t plus a derivative of tB with respect to t?

This really is hard notation for me to follow.

 

[Dick]

Just take the derivative in the middle and set t=0 on the outside factors. Another way to do it would be by writing A+tB as A(I+t*A^(-1)*B), taking the inverse and power expanding (I+t*A^(-1)*B)^(-1), but I seem to get the same thing.