Question

Solve the equation Ax = b by using the LU factorization given for A.

A= [4   -6 4         =   [1 0 0       [4 -6 4               [0

     -12 15 -7              -3 1 0        0 -3 5         b= 12

      12 -15 8]             3 -1 1]       0 0 1]             -12]

Let Ly=b and Ux=y. Solve for x and y.

y=

x=

Answer 1

Given, A =

L =

U =

b =

Let Ly = b where y = .

Then, =

i.e., y₁ = 0

- 3y₁ + y₂ = 12

3y₁ - y₂ + y₃ = -12

i.e., y₁ = 0

y₂ = 12

y₃ = 0

Therefore, = .

Again, let Ux = y where x = .

Then, =

i.e., 4x₁ - 6x₂ + 4x₃ = 0

- 3x₂ + 5x₃ = 12

x₃ = 0

i.e., x₃ = 0

x₂ = -4

x₁ = -6

Therefore, = .