# Solving matrix equation

How can I solve the forward equation

S[t]=C+A*S[t+1]*(Inverse((I-B*S(t+1)))*A


for t=1 to t=10 and S=0 with S, A, C, and B beeing 2*2 matrices?

A={{0.1,0},{0,0.1}}
B={{2,3},{-3,1}}, C={{0.2,0.6},{0.2,0}}, I={{1,0},{0,1}}

• A={{0.1,0},{0,0.1}} Mar 1 '15 at 22:32
• B={{2,3},{-3,1}}, C={{0.2,0.6},{0.2,0}}, I={{1,0},{0,1}} Mar 1 '15 at 22:33
• Please include these in the question instead of posting them as comments, such that someone can just copy all the necessary code from the question. Mar 1 '15 at 23:14

With

a = {{0.1, 0}, {0, 0.1}};
b = {{2, 3}, {-3, 1}};
c = {{0.2, 0.6}, {0.2, 0}};
i = {{1, 0}, {0, 1}};


and

s[t_] := {{s11[t], s12[t]}, {s21[t], s22[t]}}


one can see that s12 and s21 don't depend on t by evaluating

c + a*s[t + 1]*Inverse[i - b*s[t + 1]]*a


Therefore I set

s = {{0, 0.6}, {0.2, 0}}


Redefining s with

s[t_] := s[t] = c + a*s[t + 1]*Inverse[i - b*s[t + 1]]*a


one can find s for t = 10 and any other integer t <= 11 using

s

{{0.2, 0.6}, {0.2, 0.}}

• how can i define a loop for t=1 , t=10 , for above problem? Mar 2 '15 at 6:37
• @mehrnoosh You can use Table[s[t], {t, 10}]. Mar 2 '15 at 8:57
• Karsten 7. - I suspect the OP means for the recursion to be: s[t_] := s[t] = c + a.s[t + 1].Inverse[i - b.s[t + 1]].a; and the initial condition to be s={{0,0},{0,0}}. Fixing these, you can get the iteration by s[#] & /@ Range[10, 0, -1] Mar 2 '15 at 9:08
• @mehrnoosh Could you clarify if you mean Times(*, element-wise multiplication) or Dot(., matrix product). Mar 2 '15 at 10:16
a = {{a1, a2}, {a3, a4}};
b = {{b1, b2}, {b3, b4}};
c = {{c1, c2}, {c3, c4}};
s = {{0, 0}, {0, 0}};
s = c + a.s.Inverse[IdentityMatrix - b.s].a


which immediately returns

{{c1, c2}, {c3, c4}}


Trying to use upper case characters, like C, for variable names results in errors. Trying to use "abstract" vectors and matricies results in errors. Trying to use * instead of . for matrix multiplication results in errors. In slightly more complicated questions you may need to use Simplify and possibly supply assumptions about variable domains.