The solution by Cramers rule to the linear system a11x1 + a12x2 + a13x3 = b1, a21x1 +

Chapter 6, Problem 12

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The solution by Cramers rule to the linear system a11x1 + a12x2 + a13x3 = b1, a21x1 + a22x2 + a23x3 = b2, a31x1 + a32x2 + a33x3 = b3, has x1 = 1 D det b1 a12 a13 b2 a22 a23 b3 a32 a33 D1 D , x2 = 1 D det a11 b1 a13 a21 b2 a23 a31 b3 a33 D2 D , and x3 = 1 D det a11 a12 b1 a21 a22 b2 a31 a32 b3 D3 D , where D = det a11 a12 a13 a21 a22 a23 a31 a32 a33 . a. Find the solution to the linear system 2x1 + 3x2 x3 = 4, x1 2x2 + x3 = 6, x1 12x2 + 5x3 = 10, by Cramers rule. b. Show that the linear system 2x1 + 3x2 x3 = 4, x1 2x2 + x3 = 6, x1 12x2 + 5x3 = 9 does not have a solution. Compute D1, D2, and D3. c. Show that the linear system 2x1 + 3x2 x3 = 4, x1 2x2 + x3 = 6, x1 12x2 + 5x3 = 10 has an infinite number of solutions. Compute D1, D2, and D3. d. Prove that if a 3 3 linear system with D = 0 has solutions, then D1 = D2 = D3 = 0. e. Determine the number of multiplications/divisions and additions/subtractions required for Cramers rule on a 3 3 system.