Worksheet: LU Decomposition: Row Operations

In this worksheet, we will practice finding the LU decomposition (factorization) of a matrix and using the LU decomposition to simplify the solution of a system of linear equations.

Q1:

Find an LU factoring of the matrix 123213215013.

  • A 1 0 0 1 1 0 5 1 0 1 1 2 3 2 0 0 1 1 0 1 2 4 1 7
  • B 1 0 1 1 0 0 1 1 0 5 1 3 1 2 0 0 1 1 0 1 2 4 7
  • C 1 0 0 1 0 1 5 1 0 1 1 2 2 3 0 1 1 1 0 0 2 4 1 7
  • D 1 0 0 1 1 0 5 1 0 1 1 2 3 2 0 1 1 1 0 0 2 4 1 7
  • E 1 0 1 1 0 0 1 1 0 5 1 3 2 2 0 0 1 1 0 1 2 4 1 7

Q2:

Find an LU factoring of the matrix 321986622327.

  • A 1 0 0 0 3 1 0 0 2 1 1 0 1 2 2 1 3 2 1 0 2 3 0 0 1 0 0 0
  • B 1 0 0 0 3 1 0 0 2 1 1 0 1 2 2 1 3 2 1 0 2 3 0 0 1 0 0 0
  • C 1 0 0 0 3 1 0 0 2 1 1 0 1 2 2 1 2 2 1 0 3 3 0 0 1 0 0 0
  • D 1 0 0 0 3 1 0 0 2 1 1 0 1 2 2 1 3 2 1 0 2 3 0 0 1 0 0 0
  • E 1 0 0 0 3 1 0 0 2 1 1 0 1 2 2 1 3 2 1 0 2 3 0 0 1 0 0 0

Q3:

Find an LU factoring of the matrix 13113039041216.

  • A 1 0 0 0 1 1 0 0 3 0 1 0 4 0 4 1 1 3 1 0 0 1 0 0 3 0 0 0
  • B 1 0 0 0 1 1 0 0 3 0 1 0 4 0 4 1 1 3 1 0 0 1 0 0 3 0 0 0
  • C 1 0 0 0 1 1 0 0 3 0 1 0 4 0 4 1 1 3 3 0 0 2 0 0 3 0 0 0
  • D 1 0 0 0 1 1 0 0 3 0 1 0 4 0 4 1 1 3 1 0 0 1 0 0 3 0 0 0
  • E 1 0 0 0 1 1 0 0 3 0 1 0 4 0 4 1 1 3 1 0 0 1 0 0 3 0 0 0

Q4:

Find an LU factoring of the matrix 12502511336151.

  • A 1 0 0 2 0 1 3 0 1 1 2 5 0 0 1 1 3 0 0 0 1
  • B 1 0 0 2 0 1 3 0 1 1 2 5 0 0 1 1 3 0 0 0 1
  • C 1 0 0 2 1 0 3 0 1 1 2 5 0 0 1 1 3 0 0 0 1
  • D 1 0 0 2 0 1 3 0 1 1 2 2 0 0 1 1 4 0 0 0 1
  • E 1 0 0 2 0 1 3 0 1 1 2 5 0 0 0 1 3 0 0 1 0

Q5:

Find the LU decomposition for the matrix 𝐴=102411323.

  • A 1 0 0 4 1 0 2 3 1 1 0 2 0 1 9 0 0 2 8 = 1 0 2 4 1 1 3 2 3
  • B 1 0 0 2 1 0 3 2 2 1 2 0 1 0 1 3 0 0 9 2 = 1 0 2 4 1 1 3 2 3
  • C 1 0 0 4 1 0 3 2 1 1 0 2 0 1 9 0 0 2 1 = 1 0 2 4 1 1 3 2 3
  • D 1 0 0 4 1 0 3 2 1 1 0 2 0 1 9 0 0 2 1 = 1 0 2 4 1 1 3 2 3
  • E 1 0 0 4 1 0 3 2 1 1 0 2 0 1 7 0 0 8 = 1 0 2 4 1 1 3 2 3

Q6:

Find the LU decomposition for the matrix 𝐴=1011020212203211.

  • A 1 0 0 0 0 1 0 0 1 1 1 0 3 1 4 1 1 0 1 1 0 2 0 2 0 0 1 3 0 0 0 1 8 = 1 0 1 1 0 2 0 2 1 2 2 0 3 2 1 1
  • B 1 0 0 0 0 1 0 0 1 1 1 0 3 1 4 1 1 0 1 1 0 2 0 2 0 0 1 3 0 0 0 1 8 = 1 0 1 1 0 2 0 2 1 2 2 0 3 2 1 1
  • C 1 0 0 0 0 1 0 0 1 1 1 0 3 1 4 1 1 0 1 1 0 2 0 2 0 0 1 3 0 0 0 1 8 = 1 0 1 1 0 2 0 2 1 2 2 0 3 2 1 1
  • D 1 0 0 0 0 1 0 0 1 1 1 0 3 1 4 1 1 0 1 1 0 2 0 2 0 0 1 3 0 0 0 1 8 = 1 0 1 1 0 2 0 2 1 2 2 0 3 2 1 1
  • E 1 0 0 0 1 1 0 0 1 1 1 0 3 1 4 1 1 0 1 1 0 2 1 2 0 0 1 3 0 0 0 1 8 = 1 0 1 1 0 2 0 2 1 2 2 0 3 2 1 1

Q7:

Find the LU decomposition for the matrix 𝐴=183270301.

  • A 1 0 0 2 1 0 3 8 3 1 1 8 3 0 9 6 0 0 2 4 = 1 8 3 2 7 0 3 0 1
  • B 1 0 0 2 3 0 3 8 8 1 8 3 0 3 2 0 0 1 = 1 8 3 2 7 0 3 0 1
  • C 1 0 0 2 1 0 3 8 3 1 1 8 3 0 9 6 0 0 8 = 1 8 3 2 7 0 3 0 1
  • D 1 0 0 1 1 0 3 2 4 1 1 8 3 0 1 3 0 0 6 4 = 1 8 3 2 7 0 3 0 1
  • E 1 0 0 2 1 0 3 8 3 1 1 8 3 0 9 6 0 0 8 = 1 8 3 2 7 0 3 0 1

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