Worksheet: Reduced Row Echelon Form

In this worksheet, we will practice identifying the reduced row echelon form of a matrix and using Gauss-Jordan elimination to get it and hence solving a system of linear equations.

Q1:

Is the matrix 120017 in the reduced echelon form?

  • AYes
  • BNo

Q2:

Transform the matrix 120321221103 to row-reduced echelon form.

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

Q3:

Is the matrix 100000120000 in the reduced echelon form?

  • ANo
  • BYes

Q4:

Is the matrix 110005001204000013 in row-reduced echelon form?

  • AYes
  • BNo

Q5:

Reduce the matrix 121332103211 to obtain its row-reduced echelon form and then list the pivot columns.

  • A 1 0 0 0 0 1 1 0 0 1 0 1 , pivot columns: 1, 2, and 3
  • B 1 0 0 0 0 1 0 . 5 0 0 0 0 1 , pivot columns: 1, 2, and 4
  • C 1 0 0 0 0 1 1 0 0 0 0 1 , pivot columns: 1, 2, 3, and 4
  • D 1 0 0 0 0 2 1 0 0 0 0 1 , pivot columns: 1, 3, and 4
  • E 1 0 0 1 0 1 1 0 0 0 0 1 , pivot columns: 2, 3, and 4

Q6:

Reduce the matrix 123212300321 to obtain its row-reduced echelon form, and then list the pivot columns.

  • A 1 0 0 0 1 1 0 0 0 0 0 1 , pivot columns: 1 and 2
  • B 1 0 0 0 1 0 0 0 1 0 0 1 , pivot columns: 1, 2, and 3
  • C 1 0 0 0 1 0 0 0 1 0 0 0 , pivot columns: 1, 2, and 3
  • D 1 0 0 0 1 0 0 0 1 1 0 0 , pivot columns: 1, 2, and 3
  • E 1 0 0 0 1 1 1 0 1 0 0 0 , pivot columns: 1 and 2

Q7:

Give the sum of all the pivot entries in the following matrix: 𝐴=351000004001120.

Q8:

Describe the row switches which are necessary to put the following matrix into echelon form: 034000222004.

  • AWe swap rows 2 and 4, and then rows 2 and 1, and finally rows 2 and 3.
  • BWe swap rows 1 and 3, and then rows 2 and 4.
  • CWe swap rows 2 and 4, and then rows 1 and 3, and finally rows 2 and 3.
  • DWe swap rows 1 and 3, and then rows 2 and 4, and finally rows 3 and 4.
  • EThe matrix is already in echelon form; we do not need to swap any rows.

Q9:

State whether the following matrix is in echelon form: 054024002.

  • AThe matrix is not in echelon form.
  • BThe matrix is in echelon form.

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