Convex cones of matrices which are closed under matrix inversion were defined and various connections with the algebraic Lyapunov equation of general inertia were studied in (N. Cohen, I. Lewkowicz, Linear Algebra and its Appl. 250 (1997) 105-131). Here, in the price of doubling the size of the matrices involved, we introduce a unified framework for the equations of Sylvester, Lyapunov and Riccati. This enables one to extend the convex invertible cone structure to all these three equations and explore related properties. In particular, the use of the Matrix Sign Function for solving these equations is examined.