How does the active-set algorithm of fmincon use the supplied gradient information from the inequality constraints?

The constraint here is for a simplified problem compared to mine, as demonstration for my question. I have a simple, but highly non-linear constraint in reality. Consider the optimisation constraint Ax-b<=0 where A is an NxN real matrix, x and b are real N-dim vectors, with b known and x the unknown to be estimated. The Jacobian matrix of this constraint is A, containing the gradients of each individual constraint. I've supplied this matrix to fmincon and solved my problem using the active-set algorithm. I appear to get an improvement in the ability of fmincon to converge to a solution when A is invertible. Does Matlab need to invert this matrix at any point when using this algorithm? If not, is there another reason why I might see this improvement? Does anyone have a reference to literature with greater detail than the fmincon notes on how the active-set algorithm in Matlab works? I have the book 'Numerical Optimization' by J. Nocedal and S.J. Wright. Thanks.