A graph-theoretic model is introduced for bilinear algorithms. This facilitates in particular the investigation of the additive complexity of matrix multiplication. The number of additions/subtractions required for each of the problems defined by symmetric permutations on the dimensions of the matrices are shown to differ conversely as the size of each product matrix. It is noted that this result holds for any system of dual problems, not only dual matrix multiplication problems. This additive symmetry is employed to obtain various results, including the fact that 15 additive operations are necessary and sufficient to multiply two $2 \times 2$ matrices by a bilinear algorithm using at most 7 multiplication operations.