Conversely, given a basic feasible solution, the columns corresponding to the nonzero variables can be expanded to a nonsingular matrix.
From the en.wikipedia.org
Any singular covariance matrix is pivoted so that the first diagonal partition is nonsingular and well-conditioned.
From the en.wikipedia.org
The strict concavity of the objective function implies that the Jacobian of f, which is exactly the matrix of second partial derivatives of p with respect to the endogenous variables, is nonsingular.