Cramer - ME 210 Applied Mathematics for Mechanical...

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ME – 210 Applied Mathematics for Mechanical Engineers Prof. Dr. Faruk Arınç Spring 2010 Cramer’s Rule A useful application of determinants to the solution of linear algebraic equations [A] [x] = [b] is the Cramer’s rule , which can be used only when the coefficient matrix [A] is a non-singular and square matrix. Note that when [A] is a nonsingular matrix, it has a nonzero determinant (that is, det[A] ≠ 0, hence its rank is full (that is rank[A] = n), hence rank[A] = rank[A¦b], thereby the solution exists and is unique .
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ME – 210 Applied Mathematics for Mechanical Engineers Prof. Dr. Faruk Arınç Spring 2010 This unique solution is given by x i = D i / D where D is the determinant of [A] and D i is the determinant of the matrix obtained by replacing the i th column by the elements of [b]. This rule should not be used as a computational method because of the large effort required to evaluate determinants. Example: 1 2 3 1 1 2 3 2 1 2 3 3 2 x - x + 2 x = 2 2 -1 2 x 2 x + 10 x - 3 x = 5 1 10 -3 x = 5 -x + x + x = -3 -1 1 1 x -3 [x] [b] [A]  
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This note was uploaded on 07/31/2011 for the course ME 210 taught by Professor Farukarinç during the Spring '09 term at Middle East Technical University.

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Cramer - ME 210 Applied Mathematics for Mechanical...

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