This paper examines various theorems of trade and general equilibrium in a generalized framework involving arbitrary numbers of goods and factors. It develops structural relations among the changes in outputs, commodity prices, factor rewards, and factor endowments. By finding a way of inverting a bordered matrix with a singular Hessian, the paper derives explicit expressions for the following matrices: the Stolper-Samuelson matrix; the Rybczynski matrix; the matrix which measures the effect of a change in factor endowments upon factor rewards at constant commodity prices; and the matrix which measures the effect of a change in commodity prices upon outputs at constant factor endowments. Various properties of these matrices are used to obtain, among other results, the reciprocity relations and general results on factor-price equalization. The paper also examines the problem of indeterminancy in production when the number of commodities exceeds the rank of the input-coefficient matrix and presents the correct specifications of the supply functions of outputs. Finally a new theorem on the degree of flatness of the production transformation surface is derived.