Let \(X\subset {\mathbb P}(V)\) be an irreducible projective variety, where \(V\) is an \(N\) dimensional vector space over an infinite field \(K\). Then a general linear subspace \({\mathbb P}(W)\) of codimension \(\dim X+1\) will not intersect \(X\). If we start with a basis of \(V\) and write the linear equations for \(\mathbb{P}(W)\), then the equations with respect to the chosen coordinates are of the form \(\sum\lambda_{ij}X_j\) where \(\lambda_{ij}\in K\), and we can talk of the number of non-zero \(\lambda_{ij}\)'s in these equations. Of course, this is not well defined, but for example if we wrote the Plücker-coordinates, then it is. At any rate, the smallest such number we can get by choosing different \(W\) is called the complexity of \(X\). (Having a lot of zeros make \(W\) and thus \(X\) computationally simple.) This notion was introduced by \textit{D. Eisenbud} and \textit{B. Sturmfels} [J. Pure Appl. Algebra 94, No. 2, 143-157 (1994; Zbl 0807.13012)]. The authors study this concept when \(X\) is a determinantal variety -- the most ubiquitous varieties arising in computations. As one can see, this concept is closely related to the Chow form of \(X\) and the authors compute the Chow forms for certain determinantal varieties.