979 CHAPTER 9 Test Prep Concepts Examples 9.3 Determinant Solution of Linear Systems Determinant of a 2 : 2 Matrix If A = c a11 a21 a12 a22d , then ∣ A∣ = 2 a11 a21 a12 a22 2 =a11a22 −a21a12. Determinant of a 3 : 3 Matrix If A = £ a11 a21 a31 a12 a22 a32 a13 a23 a33§, then ∣ A∣ =3 a11 a21 a31 a12 a22 a32 a13 a23 a33 3 =1a11a22a33 +a12a23 a31 +a13 a21a322 − 1a31a22a13 +a32a23a11 +a33a21a122 In practice, we usually evaluate determinants by expansion by minors. Cramer’s Rule for Two Equations in Two Variables Given the system a1x + b1y = c1 a2 x + b2y = c2, if D≠0, then the system has the unique solution x = Dx D and y = Dy D , where D= 2 a1 a2 b1 b2 2 , Dx = 2 c1 c2 b1 b2 2 , and Dy = 2 a1 a2 c1 c2 2 . General Form of Cramer’s Rule Let an n * n system have linear equations of the form a1x1 + a2x2 + a3x3 + g+ anxn = b. Define D as the determinant of the n * n matrix of coefficients of the variables. Define Dx 1 as the determinant obtained from D by replacing the entries in column 1 of D with the constants of the system. Define Dxi as the determinant obtained from D by replacing the entries in column i with the constants of the system. If D≠0, then the unique solution of the system is x1 = Dx 1 D , x2 = Dx 2 D , x3 = Dx 3 D , N, xn = Dx n D . Evaluate. 2 3 -2 5 6 2 = 3162 - 1-225 = 18 + 10 = 28 Evaluate by expanding about the second column. 3 2 -1 -1 -3 -4 0 -2 -3 23 = -1-322 -1 -1 -3 22 +1-422 2 -1 -2 22 - 02 2 -1 -2 -32 = 31-52 - 4122 - 01-82 = -15 - 8 + 0 = -23 Solve using Cramer’s rule. x - 2y = -1 2x + 5y = 16 x = Dx D = 2 -1 16 -2 52 2 1 2 -2 52 = -5 + 32 5 + 4 = 27 9 = 3 y = Dy D = 2 1 2 -1 162 2 1 2 -2 52 = 16 + 2 5 + 4 = 18 9 = 2 The solution set is 513, 226. Solve using Cramer’s rule. 3x + 2y + z = -5 x - y + 3z = -5 2x + 3y + z = 0 Using the method of expansion by minors, it can be shown that Dx = 45, Dy = -30, Dz = 0, and D= -15. x = Dx D = 45 -15 = -3, y = Dy D = -30 -15 = 2, z = Dz D = 0 -15 = 0 The solution set is 51-3, 2, 026.
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