# Math Help - inverse matrix question

1. ## inverse matrix question

Is this a valid thing to do, if P and R are both square matrices whose inverse exists?:

P * inverse( I + R*P )

=

inverse( inverse(P) + R )

2. Sure:

$P(I+RP)^{-1} = \left(P^{-1}\right)^{-1}(I+RP)^{-1} = \underbrace{\left[(I+RP)P^{-1}\right]^{-1}}_{B^{-1}A^{-1} = (AB)^{-1}}$
$= \left[P^{-1} + RPP^{-1}\right]^{-1} = \left[P^{-1} + R\right]^{-1}$

3. Thank you. That ${B^{-1}A^{-1} = (AB)^{-1}}$ part was what I didn't know about.