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In this paper, the Ritz method is developed for the analysis of thin rectangular orthotropic plates undergoing large deflection. The trial functions approximating the plate lateral and in-plane displacements are represented by simple polynomials. The nonlinear algebraic equations resulting from the application of the concept of minimum potential energy of the orthotropic plate are cast in a matrix form. The developed matrix form equations are then implemented in a Mathematica code that allows for the automation of the solution for an arbitrary number of the trial polynomials. The developed code is tested through several numerical examples involving rectangular plates with different aspect ratios and boundary conditions. The results of all examples demonstrate the efficiency and accuracy of the proposed method.