The problem of finding m × s matrices (with m ≥ s) of rank r in a real affine subspace of dimension n has many applications in information and systems theory, where low rank is synonymous of structure and parsimony. We design computer algebra algorithms to solve this problem efficiently and exactly: the input are the rational coefficients of the matrices spanning the affine subspace as well as the expected maximum rank, and the output is a rational parametrization encoding a finite set of points that intersects each connected component of the low rank real algebraic set. The complexity of our algorithm is studied thoroughly. It is essentially polynomial in n+m(s−r) ; it improves on the state-of-the-art in the field. Moreover, computer experiments show the practical efficiency of our approach.