H = {{0,1,0},{0,0,1}} ;
HT = Transpose[H];

P0 = {{1,(E^((-r12)/d)),(E^((-r13)/d))},{(E^((-r12)/d)),1,(E^((-r23)/d))},{(E^((-r13)/d)),(E^((-r23)/d)),1}};

MatrixForm[P0]

H . P0 . HT//MatrixForm

Inverse[ H.P0.HT ]//MatrixForm

K=Simplify[P0 . HT . Inverse[ H.P0.HT ]];
K//MatrixForm

(* multiply by a constant to look at the simplified structure *)
(1-(E^(-((2 r23)/d)))) K//Simplify//MatrixForm

(* compute the posteriori state estimate *)
xhatPlus=K . {{p2},{p3}}//Simplify

Simplify[(1-(E^(-((2 r23)/d))))xhatPlus[[1,1]]]