We present a constant rank theorem for saddle solutions to special Lagrangian equations and quadratic Hessian equations (a minimum principle for the minimum eigenvalue of Hessian of  a solution to elliptic equations satisfying a relaxed convexity, precisely inverse-convexity condition). The argument also leads to new Liouville type results for the special Lagrangian equations with subcritical phase, matching the known rigidity results for semi-convex entire solutions to the quadratic Hessian equation. This is joint work with W. Jacob Ogden.