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A Generalization of Cramer’s Rule

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DOI: 10.4236/alamt.2015.54016    4,551 Downloads   5,956 Views   Citations
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ABSTRACT

In this paper, we find two formulas for the solutions of the following linear equation , where is a real matrix. This system has been well studied since the 1970s. It is known and simple proven that there is a solution for all if, and only if, the rows of A are linearly independent, and the minimum norm solution is given by the Moore-Penrose inverse formula, which is often denoted by ; in this case, this solution is given by . Using this formula, Cramer’s Rule and Burgstahler’s Theorem (Theorem 2), we prove the following representation for this solution

, where are the row vectors of the matrix A. To the best of our knowledge and looking in to many Linear Algebra books, there is not formula for this solution depending on determinants. Of course, this formula coincides with the one given by Cramer’s Rule when .

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Leiva, H. (2015) A Generalization of Cramer’s Rule. Advances in Linear Algebra & Matrix Theory, 5, 156-166. doi: 10.4236/alamt.2015.54016.

References

[1] Burgstahier, S. (1983) A Generalization of Cramer’s Rulle. The Two-Year College Mathematics Journal, 14, 203-205.
http://dx.doi.org/10.2307/3027088
[2] Iturriaga, E. and Leiva, H. (2007) A Necessary and Sufficient Conditions for the Controllability of Linear System in Hilbert Spaces and Applications. IMA Journal Mathematical and Information, 25, 269-280.
http://dx.doi.org/10.1093/imamci/dnm017
[3] Curtain, R.F. and Pritchard, A.J. (1978) Infinite Dimensional Linear Systems. Lecture Notes in Control and Information Sciences, Vol. 8, Springer Verlag, Berlin.

  
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