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 .
Highlights
In this paper, we find a formula depending on determinants for the solutions of the following linear equationAx =b, x ∈ IRm, b ∈ IRn, m ≥ n (1)or a1,1x1 + a1,2 x2 + + a1,m xm =b1 a2,1 x1 + a2,2 x2 + + a2,m xm =b2 . (2)an,1x1 + an,2 x2 + + an,m xm =bnif we define the column vectors =l1
Using this formula, Cramer’s Rule and Burgstahler’s Theorem (Theorem 2), we prove the following representation for this solution l1 2 + a1i b1 l2, l1 + a1i b2
We find a formula depending on determinants for the solutions of the following linear equation
Summary
We find a formula depending on determinants for the solutions of the following linear equation. (Cramer Rule 1704-1752) If A is n × n matrix with det ( A) ≠ 0 , the solution of the system (1) is given by the formula:. Dr Sylvan Burgstahler ([1]) from University of Minnesota, Duluth, where he taught for 20 years This result is given by the following Theorem: Theorem 1.2. One solution for this equation is given by the following formula:. Where A* is the transpose of A (or the conjugate transpose of A in the complex case) This solution coincides with the Cramer formula when n = m. Formed by the rows of the matrix A is lineally independent in IRm. a solution for the system (1) is given by the following formula:.
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