Appendix_SA - Appendix SA SA.1 Solution of Linear...

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1 Appendix SA Solution of Linear Simultaneous Equations SA.1 System of Linear Simultaneous Equations A system of n linear simultaneous equations may be written in the standard form as follows: a 11 x 1 + a 12 x 2 + a 13 x 3 + + a 1 n x n = B 1 a 21 x 1 + a 22 x 2 + a 23 x 3 + + a 2 n x n = B 2 a 31 x 1 + a 32 x 2 + a 33 x 3 + + a 3 n x n = B 3 a n1 x 1 + a n2 x 2 + a n3 x 3 + + a nn x n = B n (SA.1.1) where x 1 , x 2 , , x n are the unknowns, or dependent variables, whose values are to be determined in terms of the B ’s and the a ij coefficients. These coefficients, which in general could be complex numbers, may be ordered in an n × n array as follows: nn n n n n n a a a a a a a a a a a a 3 2 1 2 23 22 21 1 13 12 11 = Δ (SA.1.2) where the i and j subscripts in a ij denote, respectively, the i th row and j th column in the array. The array (SA.1.2) is known as the determinant of the set of simultaneous equations and is denoted by the symbol Δ . The determinant is a number whose value is evaluated according to certain rules.
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2 SA.2 Solution for Two Linear Simultaneous Equations Consider the two simultaneous equations: 2 2 22 1 21 1 2 12 1 11 B x a x a B x a x a = + = + (SA.2.1) By simple elimination of variables these two equations may be solved to give: 12 21 22 11 2 12 1 22 1 a a a a B a B a x = 12 21 22 11 1 21 2 11 2 a a a a B a B a x = (SA.2.2) The solutions for x 1 and x 2 may be written as the ratios of two determinants: 22 22 12 11 22 2 12 1 2 a a a a a B a B x = 22 22 12 11 2 22 1 11 2 a a a a B a B a x = (SA.2.3) The method of solving a system of linear simultaneous equations by means of determinants is known as Cramer’s rule . The solutions given by Eqs. (SA.2.2) are derived according to the following rules: Step 1. The expression for each variable is the ratio of two determinants. The determinant in the denominator is always Δ , the determinant of the set of equations as defined by (SA.1.2) . The determinant in the numerator of x 1 is obtained by replacing the coefficients of the first column in Δ , i.e. the coefficients of x 1 in the equations, by the column representing the B ’s. Similarly, the determinant in the numerator of x 2 is obtained by replacing the coefficients of the second column in Δ , i.e. the coefficients of x 2 in the equations, by the column representing the B ’s. Step 2. Each determinant expands to the expression given: 12 2 22 1 22 2 12 1 a B a B a B a B = 1 21 2 11 2 21 1 11 B B B a B a a a = 12 21 22 11 22 21 12 11 a a a a a a a a = = Δ (SA.2.4) Each product term in the expansion of the determinant is obtained by multiplying each element in a given column, or a given row, by its minor , with the appropriate sign. The minor of a given element is what remains after
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This note was uploaded on 02/17/2011 for the course EECE 210 taught by Professor Riadchedid during the Fall '07 term at American University of Beirut.

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Appendix_SA - Appendix SA SA.1 Solution of Linear...

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