The diffusion limited aggregation model (DLA) and the more general dielectric breakdown model (DBM) are solved exactly in a two-dimensional cylindrical geometry with periodic boundary conditions of width 2. Our approach follows the exact evolution of the growing interface, using the evolution matrix [formula presented] which is a temporal transfer matrix. The eigenvector of this matrix with an eigenvalue of 1 represents the system’s steady state. This yields an estimate of the fractal dimension for DLA, which is in good agreement with simulations. The same technique is used to calculate the fractal dimension for various values of [formula presented] in the more general DBM. Our exact results are very close to the approximate results found by the fixed scale transformation approach.