The solution by Cramer’s rule to the linear system a₁₁x₁ + a₁₂x₂ + a₁₃x₃ = b₁, a₂₁x₁ + a₂₂x₂ + a₂₃x₃ = b₂, a₃₁x₁ + a₃₂x₂ + a₃₃x₃ = b₃, has

And

a. Find the solution to the linear system 2x₁ + 3x₂ − x₃ = 4, x₁ − 2x₂ + x₃ = 6, x₁ − 12x₂ + 5x₃ = 10, by Cramer’s 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 Cramer’s rule on a 3 × 3 system.

Transcribed Image Text:

aj1 bi a13 azi b2 az23 b3 a33 bi a12 a13 DI 1 XI =7 det b2 a23 D' X2 = 5 det b3 a32 a33