XV. On systems of linear indeterminate equations and congruences

Abstract The theory of the solution, in positive or negative integral numbers, of systems of linear indeterminate equations, requires the consideration of rectangular matrices, the constituents of which are integral numbers. It will therefore be convenient to explain the meaning which we shall attach to certain phrases and symbols relating to such matrices. A matrix containing p constituents in every horizontal row, and q in every vertical column, is a matrix of the type q × p. We shall employ the symbol ∥ q × pA∥ or (when it is not necessary that the type of the matrix should be indicated in its symbol) the simpler symbol ∥A∥ to represent the matrix ∥ A1, 1, A1, 2,. . . . . A1, p∥ ∥ A2, 1, A2, 2,. . . . . A2, p∥ ∥ Aq, 1, Aq, 2,. . . . . . Aq, p∥ If ∥A∥ and ∥B∥ be two matrices of the same type, the equation ∥A∥ = ∥B∥ indicates that the constituents of ∥A∥ are respectively equal to the constituents of ∥B∥; whereas the equation |A|=|B| will merely express that the determinants of ∥A∥ are equal to the corresponding determinants of ∥B∥. The determinants of a matrix are, of course, the determinants of the greatest square matrices contained in it; similarly, its minor determinants of order i are the determinants of the square matrices of the type i × i that are contained in it. Matrices of the types n×(m + n) and m × (m + n) are said to be of complementary types; if ∥A∥ and ∥B∥ be two such matrices, we shall employ the equation |A| = |B| to express that each determinant of ∥A∥ is equal to that determinant of ∥B∥, by which it is multiplied in the development of the determinant of the square matrix ∥AB∥. When m and n are both uneven numbers, the signs of the determinants and |AB| and |BA| are different: this occasions a certain ambiguity of sign in the interpretation of the equation |A| = |B|, which, however, will occasion no inconvenience. If m = n, the matrices ∥A∥ and ∥B∥ are at once of the same, and of complementary types; so that, in this case, the equation |A| = |B| may stand for either of two very different sets of equations; but this also is an imperfection of the notation here employed, which it is sufficient to have pointed out. If k denote any quantity whatever, it is hardly necessary to state that the equality |A| = k × |B| implies that determinants of |A| are respectively k times the corresponding determinants of ∥B∥.