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.

  • A𝑦=5, π‘₯=6
  • B𝑦=4, π‘₯=βˆ’3
  • C𝑦=5, π‘₯=βˆ’4
  • D𝑦=βˆ’5, π‘₯=βˆ’6
  • E𝑦=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.

  • A𝑧=1, 𝑦=2, π‘₯=6
  • B𝑧=6, 𝑦=βˆ’16, π‘₯=27
  • C𝑧=βˆ’6, 𝑦=βˆ’16, π‘₯=27
  • D𝑧=6, 𝑦=16, π‘₯=27
  • E𝑧=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‘₯𝑦𝑧=3βˆ’7𝑑4βˆ’5𝑑𝑑, π‘‘βˆˆβ„
  • Bπ‘₯𝑦𝑧=3βˆ’7𝑑5βˆ’4𝑑𝑑, π‘‘βˆˆβ„
  • Cπ‘₯𝑦𝑧=3βˆ’5𝑑5βˆ’7𝑑𝑑, π‘‘βˆˆβ„
  • Dπ‘₯𝑦𝑧=7π‘‘βˆ’34βˆ’5𝑑𝑑, π‘‘βˆˆβ„
  • Eπ‘₯𝑦𝑧=7π‘‘βˆ’35π‘‘βˆ’4𝑑, π‘‘βˆˆβ„

Q4:

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

  • A1003101βˆ’311βˆ’3βˆ’4βˆ’301βˆ’210001
  • B100βˆ’3101βˆ’311βˆ’34βˆ’301βˆ’210001
  • C100βˆ’3101311βˆ’3βˆ’4βˆ’301βˆ’210001
  • D10031013113βˆ’4301210001
  • E100βˆ’3101βˆ’311βˆ’3βˆ’4βˆ’301βˆ’210001

Q5:

Find an LU factoring of the matrix 120213123.

  • A100220101120025003
  • B10021010112003βˆ’3003
  • C1002101011200βˆ’33003
  • D10021010112003βˆ’3003
  • E1001201011200βˆ’33003

Q6:

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

  • A100310211131βˆ’101520001
  • B100310βˆ’211131101520001
  • C1003102βˆ’1βˆ’1131βˆ’101520001
  • D100βˆ’310βˆ’21113βˆ’110βˆ’1520001
  • E1003102βˆ’11131βˆ’101520001

Q7:

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

  • A1001βˆ’102βˆ’111βˆ’13βˆ’101120001
  • B100βˆ’1102βˆ’111βˆ’1βˆ’3βˆ’101120001
  • C100βˆ’1102βˆ’1βˆ’11βˆ’1βˆ’3101120001
  • D1001102βˆ’111βˆ’1βˆ’3βˆ’101120001

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