NDSolve
gives an output this way:
With[{Q = IdentityMatrix[2], B = IdentityMatrix[2],
M = IdentityMatrix[2], A = IdentityMatrix[2], R = IdentityMatrix[2]},
NDSolve[{P'[t] == -P[t] . A - Transpose[A] . P[t] -
Q + (P[t] . B + M) .
Transpose[R] . (Transpose[B] . P[t] + Transpose[M]),
P[1] == IdentityMatrix[2]}, P[t], {t, 0, 1}]]
but I don't know whether it is meaningful. Unfortunately symbolic matrices and numerical matrices do not mix well together.