Part (a) :

Solve[{A11==P1+(1-P1) B11,B11==(1-P2) A11},{A11,B11}]

Solve[{A10==P1+(1-P1) B10,B10==P2 (-((P1 (-1+P2))/(P1+P2-P1 P2)))+(1-P2) A10},{A10,B10}]//Simplify

Solve[{A01==P1(-(P1/(-P1-P2+P1 P2)))+(1-P1) B01,B01==(1-P2) A01},{A01,B01}]//Simplify

Solve[{A00==P1(((P1 (P1 (((-1+P2))^2)-(-2+P2) P2))/(((P1+P2-P1 P2))^2)))+(1-P1) B00,B00==P2(-(((P1^2) (-1+P2))/(((P1+P2-P1 P2))^2))) + (1-P2) A00},{A00,B00}]//Simplify

Part (b) :

(* Code in the boundary conditions: *)
A[3,j_]:=1;
B[3,j_]:=1;
A[i_,3]:=0;
B[i_,3]:=0;

(* Code in the recurrence relationships: *)
aEqs=Table[A[i,j]==P1 A[i+1,j]+(1-P1) B[i,j],{i,0,2},{j,0,2}]//Flatten;
bEqs=Table[B[i,j]==P2 B[i,j+1]+(1-P2) A[i,j],{i,0,2},{j,0,2}]//Flatten;
aVars=Table[A[i,j],{i,0,2},{j,0,2}]//Flatten;
bVars=Table[B[i,j],{i,0,2},{j,0,2}]//Flatten;

(* Solve for all the variables simultaneously *)
Solve[Join[aEqs,bEqs],Join[aVars,bVars]]//Simplify