On extensions of Cramer’s rule for solutions of restricted linear systems. (English) Zbl 0544.15002

The author extends a method of Ben-Israel (which derives a Cramer rule for the minimum Euclidean-norm solution of a consistent linear equation) such that it becomes possible to give the unique solution of a more general consistent restricted linear system a (correspondingly modified) determinantal form. He also introduces a different approach which results in a second and more condensed Cramer rule for the solutions of the considered problem. Moreover, he presents a determinantal formula for any \(\{\) 1,2\(\}\)-generalized inverse of an \(m\times n\) matrix A, which in the case where A is nonsingular reduces to the formula \(A^{-1}=(\det A)^{-1}adj A\); here, the symbol adj A is used to denote the adjoint matrix of A.
Reviewer: N.I.Osetinski


15A06 Linear equations (linear algebraic aspects)
15A09 Theory of matrix inversion and generalized inverses
Full Text: DOI


[1] Ben-Israel A., Generalized Inverses:Theory and Applications (1974)
[2] Ben-Israel A., Linear Algebra Appl. 43 pp 223– (1982) · Zbl 0487.15004 · doi:10.1016/0024-3795(82)90255-5
[3] Chipman J. S., J. Amer. Statist. Assoc 59 pp 1078– (1964) · doi:10.1080/01621459.1964.10480751
[4] Gröbner W., Bibliographisches Institut 59 (1965)
[5] Hartung J., Vandenhoevk & Ruprecht 59 (1984)
[6] Robinson S. M., Math Mag 43 pp 94– (1970)
[7] Searle S R., Linear Models (1971)
[8] Thrall, R. M. and Lornherm, L. 1957. ”Vector Spaces and Matrices”. John Wiley & Sons.
[9] Werner H J., Methods of Operations Research 31 pp 705– (1979)
[10] Werner H. J., Linear Algebra Appl. 27 pp 141– (1979) · Zbl 0418.15018 · doi:10.1016/0024-3795(79)90036-3
[11] Werner H. J., Z. Angew. Math. Mech. 61 pp 355– (1981)
[12] Werner H. J., Linear Algebra Appl. 61 (1984)
[13] Werner, H J. ”Generalized inversion and the concept of weak bi-complementanty. research report SFB21 (1982)”. University of Bonn (submitted).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.