Worksheet: LU Decomposition: Doolittle's Method

In this worksheet, we will practice finding the LU decomposition (factorization) of a matrix using Doolittle's method.

Q1:

Find the LU factorization of the coefficient matrix, using Doolittle’s method, and use it to solve the system of equations π‘₯+2𝑦=5 and 2π‘₯+3𝑦=6.

  • A 𝑦 = 5 , π‘₯ = 6
  • B 𝑦 = 4 , π‘₯ = βˆ’ 3
  • C 𝑦 = 5 , π‘₯ = βˆ’ 4
  • D 𝑦 = βˆ’ 5 , π‘₯ = βˆ’ 6
  • E 𝑦 = βˆ’ 4 , π‘₯ = 3

Q2:

Consider the equations π‘₯+2𝑦+𝑧=1, 𝑦+3𝑧=2, and 2π‘₯+3𝑦=6. Use Doolittle’s method to find an LU factorization of the coefficient matrix of this system of equations, and hence solve the system.

  • A 𝑧 = 1 , 𝑦 = 2 , π‘₯ = 6
  • B 𝑧 = 6 , 𝑦 = βˆ’ 1 6 , π‘₯ = 2 7
  • C 𝑧 = βˆ’ 6 , 𝑦 = βˆ’ 1 6 , π‘₯ = 2 7
  • D 𝑧 = 6 , 𝑦 = 1 6 , π‘₯ = 2 7
  • E 𝑧 = 1 , 𝑦 = βˆ’ 2 , π‘₯ = 6

Q3:

Consider the following system of equations: π‘₯+2𝑦+3𝑧=5,2π‘₯+3𝑦+𝑧=6,3π‘₯+5𝑦+4𝑧=11. Use Doolittle’s method to find an LU factorization of the coefficient matrix of this system of equations, and hence solve the system.

  • A  π‘₯ 𝑦 𝑧  =  3 βˆ’ 7 𝑑 4 βˆ’ 5 𝑑 𝑑  , 𝑑 ∈ ℝ
  • B  π‘₯ 𝑦 𝑧  =  3 βˆ’ 7 𝑑 5 βˆ’ 4 𝑑 𝑑  , 𝑑 ∈ ℝ
  • C  π‘₯ 𝑦 𝑧  =  3 βˆ’ 5 𝑑 5 βˆ’ 7 𝑑 𝑑  , 𝑑 ∈ ℝ
  • D  π‘₯ 𝑦 𝑧  =  7 𝑑 βˆ’ 3 4 βˆ’ 5 𝑑 𝑑  , 𝑑 ∈ ℝ
  • E  π‘₯ 𝑦 𝑧  =  7 𝑑 βˆ’ 3 5 𝑑 βˆ’ 4 𝑑  , 𝑑 ∈ ℝ

Q4:

Find an LU factoring of the matrix 1βˆ’3βˆ’4βˆ’3βˆ’31010101βˆ’62βˆ’5.

  • A  1 0 0 3 1 0 1 βˆ’ 3 1   1 βˆ’ 3 βˆ’ 4 βˆ’ 3 0 1 βˆ’ 2 1 0 0 0 1 
  • B  1 0 0 βˆ’ 3 1 0 1 βˆ’ 3 1   1 βˆ’ 3 4 βˆ’ 3 0 1 βˆ’ 2 1 0 0 0 1 
  • C  1 0 0 βˆ’ 3 1 0 1 3 1   1 βˆ’ 3 βˆ’ 4 βˆ’ 3 0 1 βˆ’ 2 1 0 0 0 1 
  • D  1 0 0 3 1 0 1 3 1   1 3 βˆ’ 4 3 0 1 2 1 0 0 0 1 
  • E  1 0 0 βˆ’ 3 1 0 1 βˆ’ 3 1   1 βˆ’ 3 βˆ’ 4 βˆ’ 3 0 1 βˆ’ 2 1 0 0 0 1 

Q5:

Find an LU factoring of the matrix 120213123.

  • A  1 0 0 2 2 0 1 0 1   1 2 0 0 2 5 0 0 3 
  • B  1 0 0 2 1 0 1 0 1   1 2 0 0 3 βˆ’ 3 0 0 3 
  • C  1 0 0 2 1 0 1 0 1   1 2 0 0 βˆ’ 3 3 0 0 3 
  • D  1 0 0 2 1 0 1 0 1   1 2 0 0 3 βˆ’ 3 0 0 3 
  • E  1 0 0 1 2 0 1 0 1   1 2 0 0 βˆ’ 3 3 0 0 3 

Q6:

Find an LU factoring of the matrix 131βˆ’13108βˆ’125βˆ’3βˆ’3.

  • A  1 0 0 3 1 0 2 1 1   1 3 1 βˆ’ 1 0 1 5 2 0 0 0 1 
  • B  1 0 0 3 1 0 βˆ’ 2 1 1   1 3 1 1 0 1 5 2 0 0 0 1 
  • C  1 0 0 3 1 0 2 βˆ’ 1 βˆ’ 1   1 3 1 βˆ’ 1 0 1 5 2 0 0 0 1 
  • D  1 0 0 βˆ’ 3 1 0 βˆ’ 2 1 1   1 3 βˆ’ 1 1 0 βˆ’ 1 5 2 0 0 0 1 
  • E  1 0 0 3 1 0 2 βˆ’ 1 1   1 3 1 βˆ’ 1 0 1 5 2 0 0 0 1 

Q7:

Find an LU factoring of the matrix 1βˆ’1βˆ’3βˆ’1βˆ’12432βˆ’3βˆ’7βˆ’3.

  • A  1 0 0 1 βˆ’ 1 0 2 βˆ’ 1 1   1 βˆ’ 1 3 βˆ’ 1 0 1 1 2 0 0 0 1 
  • B  1 0 0 βˆ’ 1 1 0 2 βˆ’ 1 1   1 βˆ’ 1 βˆ’ 3 βˆ’ 1 0 1 1 2 0 0 0 1 
  • C  1 0 0 βˆ’ 1 1 0 2 βˆ’ 1 βˆ’ 1   1 βˆ’ 1 βˆ’ 3 1 0 1 1 2 0 0 0 1 
  • D  1 0 0 1 1 0 2 βˆ’ 1 1   1 βˆ’ 1 βˆ’ 3 βˆ’ 1 0 1 1 2 0 0 0 1 

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