A classical proof can be found here:
H. L. Royden, MR 948079 Comparison theorems for the matrix Riccati equation, Comm. Pure Appl. Math. 41 (1988), no. 5, 739--746.
Actually, more general version of this statement holds for more general version of the matrix Riccati equation. A self-contained and perhaps shorter proof, including your case, can be found in Appendix A (Thms. 38 and 40) of
D. Barilari, L. Rizzi, MR3491778 Comparison theorems for conjugate points in sub-Riemannian geometry, ESAIM Control Optim. Calc. Var. 22 (2016), no. 2, 439--472.