inverse matrix question

**Nakanja** · July 1st 2008, 09:48 AM

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 )

**o_O** · July 1st 2008, 10:13 AM

Sure:

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

**Nakanja** · July 1st 2008, 10:25 AM

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

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