It may not be the best answer.
a=[0 1;1 -1];
q=[1 0;0 1];
syms p1 p2 p3 p4 p;
p=[p1 p2;p3 p4];
f=a'*p+p.'*a+q;
pp=solve(f(1),f(2),f(3),f(4),'p1','p2','p3','p4');
[pp.p1 pp.p2;pp.p3 pp.p4]
ans =
[ -1/2-p4,-1/2+p4]
[ -1/2,p4]