MAE 3210 Homework 3 Solved

Description

1. Given the equations

10x₁ + 2x₂ − x₃ = 27

−3x₁ − 6x₂ + 2x₃ = −61.5 x₁ + x₂ + 5x₃ = −21.5,

• Solve using naive Gauss elimination (by hand). Show all steps of the computation.
• Substitute your results into the original equations to check your answers.2. Given the equations

x₁ + 2x₂ − x₃ = 2

5x₁ + 2x₂ + 2x₃ = 9

−3x₁ + 5x₂ − x₃ = 1,

• Solve by Gauss elimination with partial pivoting using code you have writtenyourself (see Figure 9.6 on page 268 of text for pseudocode – beware of typos and/or unneccessary components!).
• Substitute your results into the original equations to check your answers.

3. Given the equations

8x₁ + 4x₂ − x₃ = 11

−2x₁ + 5x₂ + x₃ = 4

2x₁ − x₂ + 6x₃ = 7,

• Solve using LU decomposition without pivoting (by hand). Show all steps ofthe computation.
• Determine the matrix inverse using LU decomposition (by hand), and verifythat [A][A]⁻¹ = [I].

4. Given the equations

2x₁ − 6x₂ − x₃ = −38

−3x₁ − x₂ + 7x₃ = −34

−8x₁ + x₂ − 2x₃ = −20,

• Solve using LU decomposition with partial pivoting using code you havewritten yourself (see Figure 10.2 on page 286 for pseudocode – beware of typos and/or unnecessary components!).
• Determine the matrix inverse using code you have written yourself (see Figure 10.5 on page 290 for pseudocode – beware of typos and/or unnecessary components!).