The problem of finding low rank m × m matrices in a real affine subspace of dimension n has many applications in information and systems theory, where low rank is synonymous of structure and parcimony. We design a symbolic computation algorithm to solve this problem efficiently, exactly and rigorously: 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 binomial(n+m(m−r),n) where r is the expected maximum rank; it improves on the state-of-the-art in the field. Moreover, computer experiments show the practical efficiency of our approach.