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A Cramer rule for finding the solution of a class of singular equations. (English) Zbl 0671.15006

Let A be an \(n\times n\) complex matrix and let k be the least nonnegative integer such that rank \(A^ k=rank A^{k+1}\). If b is in the range of \(A^ k\), then there is a unique x in the range of \(A^ k\) such that \(Ax=b\). The author obtains a Cramer’s rule type result for finding this solution. The result reduces to the usual Cramer’s rule for nonsingular A.
Reviewer: G.P.Barker

MSC:

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

[1] Robinson, S. M., A short proof of Cramer’s rule, Math. Mag.. (Montgomery, S., Selected Papers on Algebra (1977), Math. Assoc. Amer), 43, 313-314 (1970)
[2] Ben-Israel, A., A Cramer rule for least-norm solution of consistent linear equations, Linear Algebra Appl., 43, 223-226 (1982) · Zbl 0487.15004
[3] Verghese, G. C., A “Cramer rule” for least-norm least-square-error solution of inconsistent linear equations, Linear Algebra Appl., 48, 315 (1982) · Zbl 0501.15004
[4] Wang, Guo-rong, A Cramer rule for minimum-norm (T) least squares (S) solution of inconsistent linear equations, Linear Algebra Appl., 74, 213-218 (1986) · Zbl 0588.15005
[5] Campbell, S. L.; Meyer, C. D., Generalized Inverse of Linear Transformations (1979), Pitman: Pitman London · Zbl 0417.15002
[6] Ben-Israel, A.; Greville, T. N.E, Generalized Inverses: Theory and Applications (1974), Wiley-Interscience · Zbl 0305.15001
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