We introduce the polynomial coefficient matrix and identify maximum rank of this matrix under variable substitution as a complexity measure for multivariate polynomials. We use our techniques to prove super-polynomial lower bounds against several classes of non-multilinear arithmetic circuits. In particular, we obtain the following results : - As our first main result, we prove that any homogeneous depth-3 circuit for computing the product of d matrices of dimension n x n requires Ω(nd-1/2d) size. This improves the lower bounds in  for d = ω(1). - As our second main result, we show that there is an explicit polynomial on n variables and degree at most n/2 for which any depth-3 circuit C of product dimension at most n/10 (dimension of the space of affine forms feeding into each product gate) requires size 2Ω(n). This generalizes the lower bounds against diagonal circuits proved in . Diagonal circuits are of product dimension 1. - We prove a nΩ(log n) lower bound on the size of product-sparse formulas. By definition, any multilinear formula is a product-sparse formula. Thus, this result extends the known super-polynomial lower bounds on the size of multilinear formulas . - We prove a 2 Ω(n) lower bound on the size of partitioned arithmetic branching programs. This result extends the known exponential lower bound on the size of ordered arithmetic branching programs . © 2013 Springer-Verlag.