Let A_0 , A_1 , . . . , A_n be given square matrices of size m with rational coefficients. The paper focuses on the exact computation of one point in each connected component of the real determinantal variety {x ∈ R^n : det(A_0 + x_1 A_1 + · · · + x_n A_n) = 0}. Such a problem finds applications in many areas such as control theory, computational geometry, optimization, etc. Using standard complexity results this problem can be solved using m^{O(n)} arithmetic operations. Under some genericity assumptions on the coefficients of the matrices, we provide an algorithm solving this problem whose runtime is essentially quadratic in {{n+m}\choose{n}}^{3} . We also report on experiments with a computer implementation of this algorithm. Its practical performance illustrates the complexity estimates. In particular, we emphasize that for subfamilies of this problem where m is fixed, the complexity is polynomial in n.