Originally published in 1913, this book forms part of a three-volume work created to expand upon the content of a series of lectures delivered at the University of Calcutta during the winter of 1909-10. The chief feature of all three volumes is that they deal with rectangular matrices and determinoids as distinguished from square matrices and determinants, the determinoid of a rectangular matrix being related to it in the same way as a determinant is related to a square matrix. An attempt is made to set forth a complete and consistent theory or calculus of rectangular matrices and determinoids. The first volume contains the most fundamental portions of the theory and concludes with the solution of any system of linear algebraic equations, which is treated as a special case of the solution of a matrix equation of the first degree.