$[3$ points$].$ Prove the following matrix identity

$($P $1+$ BT R$1$B$)1$BT R$1=$ P BT $($BP BT $+$ R$)1,(1)$

where P in RN $$N $,$ B in RM $$N $,$ and R in RM $$M $.$ P and R are invertible.

Note that if M N $,$ it will be much cheaper to evaluate the right$-$hand side

than the left$-$hand side. Using Eq$.(1),$ prove a special case

$($I $+$ AB$)1$A $=$ A$($I $+$ BA$)1,$

where A in RN $$M and B in RM $$N