Given the equations
10x1 + 2x2 − x3 = 27
−3x1 − 6x2 + 2x3 = −61.5 x1 + x2 + 5x3 = −21.5,
(a) Solve using naive Gauss elimination (by hand). Show all steps of the computation.
(b) 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,
(a) 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!).
(b) 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,
(a) Solve using LU decomposition without pivoting (by hand). Show all steps ofthe computation.
(b) 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,
(a) 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!).
(b) 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!).
