In this paper the author is concerned with some of the ways in which the Schur complement can be used in numerical linear algebra. Variable elimination and block pivoting are first outlined: the quotient property is next used in order to test whether the leading principal minors of a matrix are nonzero or of a particular sign. Another useful application is in computing the inertia of a real symmetric matrix: the inertia is instrumental in checking such matrices for positive (semi-) definiteness. This can be exploited in mathematical programming problems in order to check a non convex quadratic function for quasi-convexity (pseudo-convexity) on the nonnegative orthant.