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1 SOLVING LINEAR ALGEBRAIC SYSTEMS and CRAMER'S RULE These notes should provide you with a brief review of the facts about linear algebraic systems and a method, Cramer's Rule, that is useful in solving small systems (no larger than 4 4). For larger systems, it is more convenient, and more accurate, to solve the system by Gaussian elimination. In order to see what the possibilities are, it is useful to look at some geometry associated with systems. If we consider such a system a 1 x + b 1 y = k 1 a 1 x + b y = k (1) then we all know that if we graph either relation on a copy of the (x; y) plane, the graph will be a pair of straight lines. Now there are three possibilities (1) the two graphs intersect, or () the two graphs are parallel, or (3) the two graphs, being really graphs of the same line, coincide. In the rst of these situations, the unique pointontersection represents the unique solution of the pair of algebraic equations. The second and third cases correspond to the situation in which there are, respectively, no solutions or innitely many solutions. Here are three concrete examples Example 1 Then the graphs look like those in Fig. 1. Example x +3y = 1; 3 x y = 3 () Then the graphs look like those in Fig.. x +3y = 1; 4 x +6y = 4 (3)

2 1 y x Figure 1 Graph of Ex. 1 1 y x Figure Graph of Ex. Example 3 See Fig. 3. x +y = 1; 3 x 6 y = 3 (4) Looking at the general case, we can use elimination to solve the

3 y x Figure 3 Graph of Ex. 3 system. The result is x = k 1b k b 1 a 1 b a b 1 ; y = a 1k a k 1 a 1 b a b 1 (5) provided that a 1 b a b 1 6= 0. If we look at our examples, we can easily check that this last relation is true only in the rst of them (1) For example 1 a 1 b a b 1 = ( ) 3 3= 13, () For example a 1 b a b 1 = 6 3 4=0, (3) For example 3 a 1 b a b 1 =1 ( 6) ( 3) =0. In the case of the rst example, we have the unique solution x = 1 ( ) = ; y = = 3 13 In the second case, the system is inconsistent. Notice that multiplication of both sides of the second equation by 1 leads to the system x +3y = 1 x +3y = ; (6) which cannot be simultaneously true for any pair of real or complex numbers (x; y). Hence there are no solutions of this system.

4 In the third case, multiplication of both sides of the second equation by the number 1 leads to the system 3 x +y = 1 x +y = 1; (7) which is not a system at all, but only a single equation in two unknowns. Looking at the geometric picture (Fig. 3) we see that all the number pairs (x; y) corresponding to points lying on the line, satisfy the equation of the line and hence represent innitely many solutions of the original solution. Exercise Analyze each of the following systems 8 < x + y =0 x y =1 x +y z =1 and 8 < 8 < x + y =0 x y =1 x +y + z =1 x + y =0 x y =1 x +y z = In the case in which there is a unique solution, nd it by elimination. (Ans.( 1 ; 1 ; 1 )). If we look again at the solution formula (5) then we can recognize the common denominator in the expressions for x and y as a determinant, namely the determinant associated with the coefcients of the left hand side of the system a 1 b 1 a b = a 1 b a b 1 Indeed, the numerators can also be recognized as determinants k 1 b 1 = k b k 1b k b 1 ;

5 in the rst case and a 1 k 1 a k = a k a k 1 ; in the second. Thus we could write the solution in terms of ratios of determinants as x = k 1 b 1 k b a 1 k 1 a k ; y = a b a b a 1 b 1 a 1 b 1 These expressions for the unknowns x and y in the general system (1) is an instance of a general rule for computing the solutions of a system of n equations in n unknowns known as Cramer's Rule. For the general system Ax = k; (8) Cramer's Rule states that the solution of the equation (11), in terms of the components x k of the solution vector x, can be written as x i = det(k i) det(a) where K i is the matrix that is obtained from the matrix A by replacing the k th column of A by the column vector k. Clearly this rule requires the evaluation of n n determinants which, if n > 4 is much longer and more inefcient than the use of Gaussian elimination. However, for and 3 3 system it is a quick way to compute the solutions of a system. Note that, as in the case, we require that the determinant of the matrix of coefcients of the unknowns is non zero, or otherwise said, we require that the system is non singular. Exercise Write down the unique solution of the 3 3 non singular example of the preceding exercise using Cramer's Rule. It only remains to give a criterion by which we can recognize, in the case that the system is singular whether the system is inconsistent or whether innitely many solutions exist. One simple criterion is usually called the Fredholm Alternative and involves the transpose A T of the system matrix A. Either the system has a unique solution or the system Ax = k A T y =0 (9)

6 has non trivial solutions and Ax = k is consistent if and only if the dot product k y =0for all solutions of A T y =0 You can check, in each of the examples above, that this alternative holds.

Section 1.1 System of Linear Equations. Dr. Abdulla Eid. College of Science. MATHS 211: Linear Algebra

Section 1.1 System of Linear Equations College of Science MATHS 211: Linear Algebra (University of Bahrain) Linear System 1 / 33 Goals:. 1 Define system of linear equations and their solutions. 2 To represent