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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 -1 K , 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 -1 K and H (adj A ) K is that in the latter case a final division of all elements by | A | is not performed.
A. C. Aitken (1938) studied this question.