XII.—Studies in Practical Mathematics. I. The Evaluation, with Applications, of a Certain Triple Product Matrix

Abstract

The solution of simultaneous linear algebraic equations, the evaluation of the adjugate or the reciprocal of a given square matrix, and the evaluation of the bilinear or quadratic form reciprocal to a given form, are all special cases of a certain general operation, namely the evaluation of a matrix product H′A-1K, where A is square and non-singular, that is, the determinant | A | is not zero. (Matrix multiplication is like determinant multiplication, but exclusively row-into-column. The matrix H′ is obtained from H by transposition, that is, by changing rows into columns.) The matrices H′ and K may be rectangular. If A is singular, the reciprocal A-1 does not exist; and in such a case the product H′(adj A) K may be required. Arithmetically, the only difference in the computation of H′A-1K and H(adj A) K is that in the latter case a final division of all elements by | A | is not performed.