E2 - Reduced row echelon form


Example 1 πŸ”—

  1. Show that

    \[\operatorname{RREF} \left[\begin{array}{ccccc} 1 & 1 & -2 & -4 & 0 \\ 0 & 1 & -1 & -3 & -3 \\ 0 & -2 & 2 & 6 & 6 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & -1 & -1 & 3 \\ 0 & 1 & -1 & -3 & -3 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .\]

  2. Explain why the matrix \(B= \left[\begin{array}{ccccc} 1 & 0 & 0 & 1 & -1 \\ 7 & 1 & -1 & 7 & -8 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] \) is or is not in reduced row echelon form.

Answer:

  1. \(\operatorname{RREF} \left[\begin{array}{ccccc} 1 & 1 & -2 & -4 & 0 \\ 0 & 1 & -1 & -3 & -3 \\ 0 & -2 & 2 & 6 & 6 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & -1 & -1 & 3 \\ 0 & 1 & -1 & -3 & -3 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .\)
  2. \(B\) is not in reduced row echelon form because not every entry above and below each pivot is zero.

Example 2 πŸ”—

  1. Show that

    \[\operatorname{RREF} \left[\begin{array}{cccc} 0 & 1 & -1 & 2 \\ -1 & 1 & 4 & -9 \\ 1 & -1 & -3 & 7 \\ 1 & -2 & -4 & 9 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & -2 \\ 0 & 0 & 0 & 0 \end{array}\right] .\]

  2. Explain why the matrix \(B= \left[\begin{array}{cccc} 1 & 0 & 7 & 22 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 \end{array}\right] \) is or is not in reduced row echelon form.

Answer:

  1. \(\operatorname{RREF} \left[\begin{array}{cccc} 0 & 1 & -1 & 2 \\ -1 & 1 & 4 & -9 \\ 1 & -1 & -3 & 7 \\ 1 & -2 & -4 & 9 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & -2 \\ 0 & 0 & 0 & 0 \end{array}\right] .\)
  2. \(B\) is not in reduced row echelon form because not every entry above and below each pivot is zero.

Example 3 πŸ”—

  1. Show that

    \[\operatorname{RREF} \left[\begin{array}{cccc} 1 & 10 & -10 & -8 \\ 1 & 2 & 1 & 9 \\ 0 & -3 & 4 & 6 \\ -1 & -6 & 1 & -11 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 \end{array}\right] .\]

  2. Explain why the matrix \(B= \left[\begin{array}{cccc} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & -2 & 2 \\ 0 & 0 & 0 & 0 \end{array}\right] \) is or is not in reduced row echelon form.

Answer:

  1. \(\operatorname{RREF} \left[\begin{array}{cccc} 1 & 10 & -10 & -8 \\ 1 & 2 & 1 & 9 \\ 0 & -3 & 4 & 6 \\ -1 & -6 & 1 & -11 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 \end{array}\right] .\)
  2. \(B\) is not in reduced row echelon form because the pivots are not all \(1\).

Example 4 πŸ”—

  1. Show that

    \[\operatorname{RREF} \left[\begin{array}{ccccc} 3 & 12 & 12 & 1 & 8 \\ -2 & -8 & -8 & 7 & 10 \\ 1 & 4 & 4 & 2 & 6 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 4 & 4 & 0 & 2 \\ 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .\]

  2. Explain why the matrix \(B= \left[\begin{array}{ccccc} 0 & 1 & 2 & 0 & 0 \\ 1 & 0 & 2 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] \) is or is not in reduced row echelon form.

Answer:

  1. \(\operatorname{RREF} \left[\begin{array}{ccccc} 3 & 12 & 12 & 1 & 8 \\ -2 & -8 & -8 & 7 & 10 \\ 1 & 4 & 4 & 2 & 6 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 4 & 4 & 0 & 2 \\ 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .\)
  2. \(B\) is not in reduced row echelon form because the pivots are not descending to the right.

Example 5 πŸ”—

  1. Show that

    \[\operatorname{RREF} \left[\begin{array}{ccccc} 1 & 4 & -2 & -5 & 4 \\ 0 & 0 & 1 & 2 & -2 \\ 0 & 0 & -3 & -6 & 6 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 4 & 0 & -1 & 0 \\ 0 & 0 & 1 & 2 & -2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .\]

  2. Explain why the matrix \(B= \left[\begin{array}{ccccc} 4 & 0 & -4 & 0 & -4 \\ 0 & 1 & 1 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] \) is or is not in reduced row echelon form.

Answer:

  1. \(\operatorname{RREF} \left[\begin{array}{ccccc} 1 & 4 & -2 & -5 & 4 \\ 0 & 0 & 1 & 2 & -2 \\ 0 & 0 & -3 & -6 & 6 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 4 & 0 & -1 & 0 \\ 0 & 0 & 1 & 2 & -2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .\)
  2. \(B\) is not in reduced row echelon form because the pivots are not all \(1\).

Example 6 πŸ”—

  1. Show that

    \[\operatorname{RREF} \left[\begin{array}{cccc} 1 & -1 & -9 & 6 \\ 0 & 1 & 4 & -3 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 4 & -6 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -5 & 0 \\ 0 & 1 & 4 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right] .\]

  2. Explain why the matrix \(B= \left[\begin{array}{cccc} 1 & 0 & 0 & -2 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -4 & -4 \\ 0 & 0 & 0 & 0 \end{array}\right] \) is or is not in reduced row echelon form.

Answer:

  1. \(\operatorname{RREF} \left[\begin{array}{cccc} 1 & -1 & -9 & 6 \\ 0 & 1 & 4 & -3 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 4 & -6 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -5 & 0 \\ 0 & 1 & 4 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right] .\)
  2. \(B\) is not in reduced row echelon form because the pivots are not all \(1\).

Example 7 πŸ”—

  1. Show that

    \[\operatorname{RREF} \left[\begin{array}{ccc} -8 & 4 & 4 \\ -1 & 3 & 8 \\ -3 & 2 & 3 \\ 1 & -2 & -5 \\ -2 & 4 & 10 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & 3 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .\]

  2. Explain why the matrix \(B= \left[\begin{array}{ccc} 1 & 0 & 1 \\ 2 & 1 & 3 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \) is or is not in reduced row echelon form.

Answer:

  1. \(\operatorname{RREF} \left[\begin{array}{ccc} -8 & 4 & 4 \\ -1 & 3 & 8 \\ -3 & 2 & 3 \\ 1 & -2 & -5 \\ -2 & 4 & 10 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & 3 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .\)
  2. \(B\) is not in reduced row echelon form because not every entry above and below each pivot is zero.

Example 8 πŸ”—

  1. Show that

    \[\operatorname{RREF} \left[\begin{array}{ccc} -2 & -2 & -2 \\ 0 & 1 & 2 \\ 0 & -2 & -4 \\ 0 & 4 & 8 \\ 1 & -1 & -3 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & -1 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .\]

  2. Explain why the matrix \(B= \left[\begin{array}{ccc} 1 & 4 & -4 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \) is or is not in reduced row echelon form.

Answer:

  1. \(\operatorname{RREF} \left[\begin{array}{ccc} -2 & -2 & -2 \\ 0 & 1 & 2 \\ 0 & -2 & -4 \\ 0 & 4 & 8 \\ 1 & -1 & -3 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & -1 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .\)
  2. \(B\) is not in reduced row echelon form because not every entry above and below each pivot is zero.

Example 9 πŸ”—

  1. Show that

    \[\operatorname{RREF} \left[\begin{array}{cccc} 1 & 0 & 1 & 1 \\ 0 & 1 & 5 & -11 \\ -4 & -1 & -8 & 5 \\ 5 & 1 & 6 & 2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 3 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & -2 \\ 0 & 0 & 0 & 0 \end{array}\right] .\]

  2. Explain why the matrix \(B= \left[\begin{array}{cccc} 0 & 0 & 1 & 2 \\ 0 & 1 & 0 & 2 \\ 1 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 \end{array}\right] \) is or is not in reduced row echelon form.

Answer:

  1. \(\operatorname{RREF} \left[\begin{array}{cccc} 1 & 0 & 1 & 1 \\ 0 & 1 & 5 & -11 \\ -4 & -1 & -8 & 5 \\ 5 & 1 & 6 & 2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 3 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & -2 \\ 0 & 0 & 0 & 0 \end{array}\right] .\)
  2. \(B\) is not in reduced row echelon form because the pivots are not descending to the right.

Example 10 πŸ”—

  1. Show that

    \[\operatorname{RREF} \left[\begin{array}{cccc} 1 & 3 & 10 & 6 \\ 0 & 1 & 3 & 2 \\ 0 & -1 & -3 & -2 \\ -1 & -2 & -7 & -4 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 1 & 0 \\ 0 & 1 & 3 & 2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] .\]

  2. Explain why the matrix \(B= \left[\begin{array}{cccc} 1 & 0 & 1 & -2 \\ -2 & 1 & -2 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \) is or is not in reduced row echelon form.

Answer:

  1. \(\operatorname{RREF} \left[\begin{array}{cccc} 1 & 3 & 10 & 6 \\ 0 & 1 & 3 & 2 \\ 0 & -1 & -3 & -2 \\ -1 & -2 & -7 & -4 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 1 & 0 \\ 0 & 1 & 3 & 2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] .\)
  2. \(B\) is not in reduced row echelon form because not every entry above and below each pivot is zero.

Example 11 πŸ”—

  1. Show that

    \[\operatorname{RREF} \left[\begin{array}{ccccc} 1 & -1 & 2 & 3 & 2 \\ -2 & 3 & -4 & -8 & -6 \\ 5 & 0 & 10 & 5 & 0 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 2 & 1 & 0 \\ 0 & 1 & 0 & -2 & -2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .\]

  2. Explain why the matrix \(B= \left[\begin{array}{ccccc} 1 & 5 & 0 & -3 & 2 \\ 0 & 0 & -6 & -12 & 6 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] \) is or is not in reduced row echelon form.

Answer:

  1. \(\operatorname{RREF} \left[\begin{array}{ccccc} 1 & -1 & 2 & 3 & 2 \\ -2 & 3 & -4 & -8 & -6 \\ 5 & 0 & 10 & 5 & 0 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 2 & 1 & 0 \\ 0 & 1 & 0 & -2 & -2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .\)
  2. \(B\) is not in reduced row echelon form because the pivots are not all \(1\).

Example 12 πŸ”—

  1. Show that

    \[\operatorname{RREF} \left[\begin{array}{ccc} 1 & 0 & 3 \\ -1 & 1 & -6 \\ 3 & 4 & -3 \\ -1 & -4 & 9 \\ -1 & -1 & 0 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 3 \\ 0 & 1 & -3 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .\]

  2. Explain why the matrix \(B= \left[\begin{array}{ccc} 0 & 1 & 0 \\ 1 & 0 & -2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \) is or is not in reduced row echelon form.

Answer:

  1. \(\operatorname{RREF} \left[\begin{array}{ccc} 1 & 0 & 3 \\ -1 & 1 & -6 \\ 3 & 4 & -3 \\ -1 & -4 & 9 \\ -1 & -1 & 0 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 3 \\ 0 & 1 & -3 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .\)
  2. \(B\) is not in reduced row echelon form because the pivots are not descending to the right.

Example 13 πŸ”—

  1. Show that

    \[\operatorname{RREF} \left[\begin{array}{ccc} -2 & 3 & 6 \\ 3 & 10 & -9 \\ -4 & -11 & 12 \\ -1 & 0 & 3 \\ -3 & -10 & 9 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & -3 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .\]

  2. Explain why the matrix \(B= \left[\begin{array}{ccc} 1 & 0 & -1 \\ 0 & -4 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \) is or is not in reduced row echelon form.

Answer:

  1. \(\operatorname{RREF} \left[\begin{array}{ccc} -2 & 3 & 6 \\ 3 & 10 & -9 \\ -4 & -11 & 12 \\ -1 & 0 & 3 \\ -3 & -10 & 9 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & -3 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .\)
  2. \(B\) is not in reduced row echelon form because the pivots are not all \(1\).

Example 14 πŸ”—

  1. Show that

    \[\operatorname{RREF} \left[\begin{array}{ccc} -2 & 8 & -6 \\ -1 & 5 & -4 \\ 3 & -7 & 4 \\ 0 & 4 & -4 \\ 1 & -4 & 3 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & -1 \\ 0 & 1 & -1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .\]

  2. Explain why the matrix \(B= \left[\begin{array}{ccc} -6 & 0 & -6 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \) is or is not in reduced row echelon form.

Answer:

  1. \(\operatorname{RREF} \left[\begin{array}{ccc} -2 & 8 & -6 \\ -1 & 5 & -4 \\ 3 & -7 & 4 \\ 0 & 4 & -4 \\ 1 & -4 & 3 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & -1 \\ 0 & 1 & -1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .\)
  2. \(B\) is not in reduced row echelon form because the pivots are not all \(1\).

Example 15 πŸ”—

  1. Show that

    \[\operatorname{RREF} \left[\begin{array}{ccccc} 1 & -5 & 3 & -2 & 4 \\ 0 & 0 & 1 & -1 & 1 \\ 0 & 0 & 5 & -5 & 5 \end{array}\right] = \left[\begin{array}{ccccc} 1 & -5 & 0 & 1 & 1 \\ 0 & 0 & 1 & -1 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .\]

  2. Explain why the matrix \(B= \left[\begin{array}{ccccc} 1 & -4 & 2 & 9 & -9 \\ 0 & 1 & 0 & -2 & 2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] \) is or is not in reduced row echelon form.

Answer:

  1. \(\operatorname{RREF} \left[\begin{array}{ccccc} 1 & -5 & 3 & -2 & 4 \\ 0 & 0 & 1 & -1 & 1 \\ 0 & 0 & 5 & -5 & 5 \end{array}\right] = \left[\begin{array}{ccccc} 1 & -5 & 0 & 1 & 1 \\ 0 & 0 & 1 & -1 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .\)
  2. \(B\) is not in reduced row echelon form because not every entry above and below each pivot is zero.

Example 16 πŸ”—

  1. Show that

    \[\operatorname{RREF} \left[\begin{array}{cccc} 1 & 6 & 11 & 10 \\ -2 & -12 & -3 & -1 \\ 0 & 0 & -3 & -3 \\ -2 & -12 & -8 & -6 \end{array}\right] = \left[\begin{array}{cccc} 1 & 6 & 0 & -1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] .\]

  2. Explain why the matrix \(B= \left[\begin{array}{cccc} 0 & 0 & 1 & 1 \\ 1 & -4 & 0 & -2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \) is or is not in reduced row echelon form.

Answer:

  1. \(\operatorname{RREF} \left[\begin{array}{cccc} 1 & 6 & 11 & 10 \\ -2 & -12 & -3 & -1 \\ 0 & 0 & -3 & -3 \\ -2 & -12 & -8 & -6 \end{array}\right] = \left[\begin{array}{cccc} 1 & 6 & 0 & -1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] .\)
  2. \(B\) is not in reduced row echelon form because the pivots are not descending to the right.

Example 17 πŸ”—

  1. Show that

    \[\operatorname{RREF} \left[\begin{array}{ccc} -1 & -2 & -5 \\ 0 & 1 & 2 \\ -3 & 5 & 7 \\ -2 & 4 & 6 \\ -2 & 1 & 0 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .\]

  2. Explain why the matrix \(B= \left[\begin{array}{ccc} 0 & 1 & 1 \\ 1 & 0 & -1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \) is or is not in reduced row echelon form.

Answer:

  1. \(\operatorname{RREF} \left[\begin{array}{ccc} -1 & -2 & -5 \\ 0 & 1 & 2 \\ -3 & 5 & 7 \\ -2 & 4 & 6 \\ -2 & 1 & 0 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .\)
  2. \(B\) is not in reduced row echelon form because the pivots are not descending to the right.

Example 18 πŸ”—

  1. Show that

    \[\operatorname{RREF} \left[\begin{array}{ccc} 0 & -1 & -2 \\ 1 & -1 & -4 \\ 1 & -1 & -4 \\ 1 & -2 & -6 \\ -2 & 4 & 12 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & -2 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .\]

  2. Explain why the matrix \(B= \left[\begin{array}{ccc} 1 & 4 & 0 \\ 0 & 0 & -3 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \) is or is not in reduced row echelon form.

Answer:

  1. \(\operatorname{RREF} \left[\begin{array}{ccc} 0 & -1 & -2 \\ 1 & -1 & -4 \\ 1 & -1 & -4 \\ 1 & -2 & -6 \\ -2 & 4 & 12 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & -2 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .\)
  2. \(B\) is not in reduced row echelon form because the pivots are not all \(1\).

Example 19 πŸ”—

  1. Show that

    \[\operatorname{RREF} \left[\begin{array}{ccccc} 1 & 1 & 1 & 0 & 4 \\ 0 & 1 & 2 & 2 & 2 \\ 0 & 1 & 2 & 2 & 2 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & -1 & -2 & 2 \\ 0 & 1 & 2 & 2 & 2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .\]

  2. Explain why the matrix \(B= \left[\begin{array}{ccccc} 1 & -5 & 0 & -3 & -3 \\ 0 & 0 & -2 & 2 & -2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] \) is or is not in reduced row echelon form.

Answer:

  1. \(\operatorname{RREF} \left[\begin{array}{ccccc} 1 & 1 & 1 & 0 & 4 \\ 0 & 1 & 2 & 2 & 2 \\ 0 & 1 & 2 & 2 & 2 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & -1 & -2 & 2 \\ 0 & 1 & 2 & 2 & 2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .\)
  2. \(B\) is not in reduced row echelon form because the pivots are not all \(1\).

Example 20 πŸ”—

  1. Show that

    \[\operatorname{RREF} \left[\begin{array}{ccc} -2 & 5 & 7 \\ -2 & -9 & -7 \\ -1 & -3 & -2 \\ -1 & 0 & 1 \\ 0 & -3 & -3 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & -1 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .\]

  2. Explain why the matrix \(B= \left[\begin{array}{ccc} 1 & 0 & 2 \\ -5 & 1 & -8 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \) is or is not in reduced row echelon form.

Answer:

  1. \(\operatorname{RREF} \left[\begin{array}{ccc} -2 & 5 & 7 \\ -2 & -9 & -7 \\ -1 & -3 & -2 \\ -1 & 0 & 1 \\ 0 & -3 & -3 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & -1 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .\)
  2. \(B\) is not in reduced row echelon form because not every entry above and below each pivot is zero.

Example 21 πŸ”—

  1. Show that

    \[\operatorname{RREF} \left[\begin{array}{ccccc} 0 & -1 & 0 & 1 & -1 \\ 1 & 4 & 3 & -5 & 4 \\ 2 & 7 & 6 & -9 & 7 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 3 & -1 & 0 \\ 0 & 1 & 0 & -1 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .\]

  2. Explain why the matrix \(B= \left[\begin{array}{ccccc} 7 & 0 & 21 & -14 & 14 \\ 0 & 1 & 1 & -1 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] \) is or is not in reduced row echelon form.

Answer:

  1. \(\operatorname{RREF} \left[\begin{array}{ccccc} 0 & -1 & 0 & 1 & -1 \\ 1 & 4 & 3 & -5 & 4 \\ 2 & 7 & 6 & -9 & 7 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 3 & -1 & 0 \\ 0 & 1 & 0 & -1 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .\)
  2. \(B\) is not in reduced row echelon form because the pivots are not all \(1\).

Example 22 πŸ”—

  1. Show that

    \[\operatorname{RREF} \left[\begin{array}{ccccc} 1 & -4 & 5 & -6 & 10 \\ 0 & 0 & 0 & 1 & -2 \\ 2 & -8 & 10 & -7 & 10 \end{array}\right] = \left[\begin{array}{ccccc} 1 & -4 & 5 & 0 & -2 \\ 0 & 0 & 0 & 1 & -2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .\]

  2. Explain why the matrix \(B= \left[\begin{array}{ccccc} 1 & 0 & -1 & -2 & 1 \\ -7 & 1 & 7 & 14 & -8 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] \) is or is not in reduced row echelon form.

Answer:

  1. \(\operatorname{RREF} \left[\begin{array}{ccccc} 1 & -4 & 5 & -6 & 10 \\ 0 & 0 & 0 & 1 & -2 \\ 2 & -8 & 10 & -7 & 10 \end{array}\right] = \left[\begin{array}{ccccc} 1 & -4 & 5 & 0 & -2 \\ 0 & 0 & 0 & 1 & -2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .\)
  2. \(B\) is not in reduced row echelon form because not every entry above and below each pivot is zero.

Example 23 πŸ”—

  1. Show that

    \[\operatorname{RREF} \left[\begin{array}{ccc} 1 & -5 & 12 \\ 0 & 1 & -2 \\ -1 & 3 & -8 \\ 0 & 1 & -2 \\ 1 & -4 & 10 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 2 \\ 0 & 1 & -2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .\]

  2. Explain why the matrix \(B= \left[\begin{array}{ccc} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \) is or is not in reduced row echelon form.

Answer:

  1. \(\operatorname{RREF} \left[\begin{array}{ccc} 1 & -5 & 12 \\ 0 & 1 & -2 \\ -1 & 3 & -8 \\ 0 & 1 & -2 \\ 1 & -4 & 10 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 2 \\ 0 & 1 & -2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .\)
  2. \(B\) is not in reduced row echelon form because the pivots are not descending to the right.

Example 24 πŸ”—

  1. Show that

    \[\operatorname{RREF} \left[\begin{array}{ccccc} -4 & 1 & 6 & 8 & 3 \\ -3 & 4 & 11 & 6 & -1 \\ -5 & 2 & 9 & 10 & 3 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & -1 & -2 & -1 \\ 0 & 1 & 2 & 0 & -1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .\]

  2. Explain why the matrix \(B= \left[\begin{array}{ccccc} 0 & 1 & 1 & -1 & -2 \\ 1 & 0 & -3 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] \) is or is not in reduced row echelon form.

Answer:

  1. \(\operatorname{RREF} \left[\begin{array}{ccccc} -4 & 1 & 6 & 8 & 3 \\ -3 & 4 & 11 & 6 & -1 \\ -5 & 2 & 9 & 10 & 3 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & -1 & -2 & -1 \\ 0 & 1 & 2 & 0 & -1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .\)
  2. \(B\) is not in reduced row echelon form because the pivots are not descending to the right.

Example 25 πŸ”—

  1. Show that

    \[\operatorname{RREF} \left[\begin{array}{cccc} -2 & 10 & -3 & -9 \\ -1 & 5 & 4 & 1 \\ -1 & 5 & -1 & -4 \\ 1 & -5 & 0 & 3 \end{array}\right] = \left[\begin{array}{cccc} 1 & -5 & 0 & 3 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] .\]

  2. Explain why the matrix \(B= \left[\begin{array}{cccc} 0 & 1 & -2 & 4 \\ 1 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \) is or is not in reduced row echelon form.

Answer:

  1. \(\operatorname{RREF} \left[\begin{array}{cccc} -2 & 10 & -3 & -9 \\ -1 & 5 & 4 & 1 \\ -1 & 5 & -1 & -4 \\ 1 & -5 & 0 & 3 \end{array}\right] = \left[\begin{array}{cccc} 1 & -5 & 0 & 3 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] .\)
  2. \(B\) is not in reduced row echelon form because the pivots are not descending to the right.

Example 26 πŸ”—

  1. Show that

    \[\operatorname{RREF} \left[\begin{array}{ccc} 1 & -7 & -7 \\ 2 & 11 & 11 \\ 3 & 11 & 11 \\ 0 & -4 & -4 \\ 2 & 7 & 7 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .\]

  2. Explain why the matrix \(B= \left[\begin{array}{ccc} 0 & 1 & 0 \\ 1 & 0 & 2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \) is or is not in reduced row echelon form.

Answer:

  1. \(\operatorname{RREF} \left[\begin{array}{ccc} 1 & -7 & -7 \\ 2 & 11 & 11 \\ 3 & 11 & 11 \\ 0 & -4 & -4 \\ 2 & 7 & 7 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .\)
  2. \(B\) is not in reduced row echelon form because the pivots are not descending to the right.

Example 27 πŸ”—

  1. Show that

    \[\operatorname{RREF} \left[\begin{array}{ccccc} -9 & 3 & -3 & -12 & 0 \\ -2 & 3 & -3 & 2 & 7 \\ 5 & -2 & 2 & 6 & -1 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 0 & 2 & 1 \\ 0 & 1 & -1 & 2 & 3 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .\]

  2. Explain why the matrix \(B= \left[\begin{array}{ccccc} 1 & 0 & 1 & 0 & 1 \\ 0 & -4 & 0 & -8 & 4 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] \) is or is not in reduced row echelon form.

Answer:

  1. \(\operatorname{RREF} \left[\begin{array}{ccccc} -9 & 3 & -3 & -12 & 0 \\ -2 & 3 & -3 & 2 & 7 \\ 5 & -2 & 2 & 6 & -1 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 0 & 2 & 1 \\ 0 & 1 & -1 & 2 & 3 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .\)
  2. \(B\) is not in reduced row echelon form because the pivots are not all \(1\).

Example 28 πŸ”—

  1. Show that

    \[\operatorname{RREF} \left[\begin{array}{cccc} 1 & 1 & 2 & -7 \\ 0 & 1 & 4 & -9 \\ 0 & -1 & -3 & 7 \\ 2 & 0 & -7 & 10 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -2 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & -2 \\ 0 & 0 & 0 & 0 \end{array}\right] .\]

  2. Explain why the matrix \(B= \left[\begin{array}{cccc} 1 & 0 & 7 & 22 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 \end{array}\right] \) is or is not in reduced row echelon form.

Answer:

  1. \(\operatorname{RREF} \left[\begin{array}{cccc} 1 & 1 & 2 & -7 \\ 0 & 1 & 4 & -9 \\ 0 & -1 & -3 & 7 \\ 2 & 0 & -7 & 10 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -2 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & -2 \\ 0 & 0 & 0 & 0 \end{array}\right] .\)
  2. \(B\) is not in reduced row echelon form because not every entry above and below each pivot is zero.

Example 29 πŸ”—

  1. Show that

    \[\operatorname{RREF} \left[\begin{array}{ccc} 1 & -2 & 2 \\ -4 & 9 & -9 \\ -4 & 8 & -8 \\ -5 & 10 & -10 \\ -5 & 12 & -12 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & -1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .\]

  2. Explain why the matrix \(B= \left[\begin{array}{ccc} 1 & 0 & 0 \\ -4 & 1 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \) is or is not in reduced row echelon form.

Answer:

  1. \(\operatorname{RREF} \left[\begin{array}{ccc} 1 & -2 & 2 \\ -4 & 9 & -9 \\ -4 & 8 & -8 \\ -5 & 10 & -10 \\ -5 & 12 & -12 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & -1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .\)
  2. \(B\) is not in reduced row echelon form because not every entry above and below each pivot is zero.

Example 30 πŸ”—

  1. Show that

    \[\operatorname{RREF} \left[\begin{array}{ccccc} 1 & -4 & 1 & -3 & 4 \\ -1 & 4 & 0 & 0 & -1 \\ 1 & -4 & 1 & -3 & 4 \end{array}\right] = \left[\begin{array}{ccccc} 1 & -4 & 0 & 0 & 1 \\ 0 & 0 & 1 & -3 & 3 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .\]

  2. Explain why the matrix \(B= \left[\begin{array}{ccccc} 1 & 0 & 2 & 2 & -1 \\ -2 & 1 & -1 & -3 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] \) is or is not in reduced row echelon form.

Answer:

  1. \(\operatorname{RREF} \left[\begin{array}{ccccc} 1 & -4 & 1 & -3 & 4 \\ -1 & 4 & 0 & 0 & -1 \\ 1 & -4 & 1 & -3 & 4 \end{array}\right] = \left[\begin{array}{ccccc} 1 & -4 & 0 & 0 & 1 \\ 0 & 0 & 1 & -3 & 3 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .\)
  2. \(B\) is not in reduced row echelon form because not every entry above and below each pivot is zero.

Example 31 πŸ”—

  1. Show that

    \[\operatorname{RREF} \left[\begin{array}{cccc} 0 & 5 & 11 & 12 \\ -2 & -5 & -6 & 2 \\ -1 & 2 & 7 & 12 \\ -2 & -4 & -3 & 6 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -2 \\ 0 & 1 & 0 & -2 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 \end{array}\right] .\]

  2. Explain why the matrix \(B= \left[\begin{array}{cccc} 1 & 0 & 0 & 3 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 3 & 3 \\ 0 & 0 & 0 & 0 \end{array}\right] \) is or is not in reduced row echelon form.

Answer:

  1. \(\operatorname{RREF} \left[\begin{array}{cccc} 0 & 5 & 11 & 12 \\ -2 & -5 & -6 & 2 \\ -1 & 2 & 7 & 12 \\ -2 & -4 & -3 & 6 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -2 \\ 0 & 1 & 0 & -2 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 \end{array}\right] .\)
  2. \(B\) is not in reduced row echelon form because the pivots are not all \(1\).

Example 32 πŸ”—

  1. Show that

    \[\operatorname{RREF} \left[\begin{array}{ccccc} 1 & -5 & 12 & 7 & 12 \\ 0 & 1 & -2 & -1 & -2 \\ -1 & 4 & -10 & -6 & -10 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 2 & 2 & 2 \\ 0 & 1 & -2 & -1 & -2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .\]

  2. Explain why the matrix \(B= \left[\begin{array}{ccccc} 1 & 0 & 1 & 1 & -1 \\ -3 & 1 & -5 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] \) is or is not in reduced row echelon form.

Answer:

  1. \(\operatorname{RREF} \left[\begin{array}{ccccc} 1 & -5 & 12 & 7 & 12 \\ 0 & 1 & -2 & -1 & -2 \\ -1 & 4 & -10 & -6 & -10 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 2 & 2 & 2 \\ 0 & 1 & -2 & -1 & -2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .\)
  2. \(B\) is not in reduced row echelon form because not every entry above and below each pivot is zero.

Example 33 πŸ”—

  1. Show that

    \[\operatorname{RREF} \left[\begin{array}{ccc} 1 & -1 & 3 \\ 2 & -1 & 3 \\ 3 & -2 & 6 \\ 4 & -1 & 3 \\ -4 & 2 & -6 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & -3 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .\]

  2. Explain why the matrix \(B= \left[\begin{array}{ccc} 1 & 3 & -4 \\ 0 & 1 & -2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \) is or is not in reduced row echelon form.

Answer:

  1. \(\operatorname{RREF} \left[\begin{array}{ccc} 1 & -1 & 3 \\ 2 & -1 & 3 \\ 3 & -2 & 6 \\ 4 & -1 & 3 \\ -4 & 2 & -6 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & -3 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .\)
  2. \(B\) is not in reduced row echelon form because not every entry above and below each pivot is zero.

Example 34 πŸ”—

  1. Show that

    \[\operatorname{RREF} \left[\begin{array}{ccccc} 1 & 4 & 5 & -7 & 6 \\ -1 & -3 & -3 & 5 & -5 \\ -2 & -4 & -2 & 6 & -8 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & -3 & 1 & 2 \\ 0 & 1 & 2 & -2 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .\]

  2. Explain why the matrix \(B= \left[\begin{array}{ccccc} 1 & 3 & 1 & -11 & 4 \\ 0 & 1 & 0 & -3 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] \) is or is not in reduced row echelon form.

Answer:

  1. \(\operatorname{RREF} \left[\begin{array}{ccccc} 1 & 4 & 5 & -7 & 6 \\ -1 & -3 & -3 & 5 & -5 \\ -2 & -4 & -2 & 6 & -8 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & -3 & 1 & 2 \\ 0 & 1 & 2 & -2 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .\)
  2. \(B\) is not in reduced row echelon form because not every entry above and below each pivot is zero.

Example 35 πŸ”—

  1. Show that

    \[\operatorname{RREF} \left[\begin{array}{cccc} -2 & -4 & 12 & -12 \\ -1 & -4 & 10 & -10 \\ -1 & -3 & 8 & -8 \\ 0 & -2 & 4 & -4 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -2 & 2 \\ 0 & 1 & -2 & 2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] .\]

  2. Explain why the matrix \(B= \left[\begin{array}{cccc} 0 & 1 & 3 & -1 \\ 1 & 0 & -2 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \) is or is not in reduced row echelon form.

Answer:

  1. \(\operatorname{RREF} \left[\begin{array}{cccc} -2 & -4 & 12 & -12 \\ -1 & -4 & 10 & -10 \\ -1 & -3 & 8 & -8 \\ 0 & -2 & 4 & -4 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -2 & 2 \\ 0 & 1 & -2 & 2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] .\)
  2. \(B\) is not in reduced row echelon form because the pivots are not descending to the right.

Example 36 πŸ”—

  1. Show that

    \[\operatorname{RREF} \left[\begin{array}{cccc} 1 & -1 & -1 & -1 \\ 2 & -1 & 1 & 1 \\ -4 & 2 & -1 & -1 \\ 0 & -3 & -10 & -10 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right] .\]

  2. Explain why the matrix \(B= \left[\begin{array}{cccc} 1 & 0 & 0 & -2 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & -6 & -12 \\ 0 & 0 & 0 & 0 \end{array}\right] \) is or is not in reduced row echelon form.

Answer:

  1. \(\operatorname{RREF} \left[\begin{array}{cccc} 1 & -1 & -1 & -1 \\ 2 & -1 & 1 & 1 \\ -4 & 2 & -1 & -1 \\ 0 & -3 & -10 & -10 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right] .\)
  2. \(B\) is not in reduced row echelon form because the pivots are not all \(1\).

Example 37 πŸ”—

  1. Show that

    \[\operatorname{RREF} \left[\begin{array}{cccc} -3 & 3 & 6 & 3 \\ -1 & 1 & 2 & 1 \\ 2 & -3 & -6 & 0 \\ -4 & 3 & 6 & 6 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -3 \\ 0 & 1 & 2 & -2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] .\]

  2. Explain why the matrix \(B= \left[\begin{array}{cccc} 0 & 0 & 1 & 1 \\ 1 & 3 & 0 & -3 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \) is or is not in reduced row echelon form.

Answer:

  1. \(\operatorname{RREF} \left[\begin{array}{cccc} -3 & 3 & 6 & 3 \\ -1 & 1 & 2 & 1 \\ 2 & -3 & -6 & 0 \\ -4 & 3 & 6 & 6 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -3 \\ 0 & 1 & 2 & -2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] .\)
  2. \(B\) is not in reduced row echelon form because the pivots are not descending to the right.

Example 38 πŸ”—

  1. Show that

    \[\operatorname{RREF} \left[\begin{array}{ccc} -1 & -7 & 11 \\ 1 & 6 & -9 \\ -1 & -1 & -1 \\ 1 & 4 & -5 \\ 3 & 10 & -11 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 3 \\ 0 & 1 & -2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .\]

  2. Explain why the matrix \(B= \left[\begin{array}{ccc} 1 & -5 & 15 \\ 0 & 1 & -3 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \) is or is not in reduced row echelon form.

Answer:

  1. \(\operatorname{RREF} \left[\begin{array}{ccc} -1 & -7 & 11 \\ 1 & 6 & -9 \\ -1 & -1 & -1 \\ 1 & 4 & -5 \\ 3 & 10 & -11 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 3 \\ 0 & 1 & -2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .\)
  2. \(B\) is not in reduced row echelon form because not every entry above and below each pivot is zero.

Example 39 πŸ”—

  1. Show that

    \[\operatorname{RREF} \left[\begin{array}{ccccc} 0 & -1 & 1 & -1 & -2 \\ 1 & -3 & 1 & -5 & -6 \\ 0 & 3 & -3 & 3 & 6 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & -2 & -2 & 0 \\ 0 & 1 & -1 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .\]

  2. Explain why the matrix \(B= \left[\begin{array}{ccccc} 0 & 1 & -2 & 0 & 1 \\ 1 & 0 & 3 & 2 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] \) is or is not in reduced row echelon form.

Answer:

  1. \(\operatorname{RREF} \left[\begin{array}{ccccc} 0 & -1 & 1 & -1 & -2 \\ 1 & -3 & 1 & -5 & -6 \\ 0 & 3 & -3 & 3 & 6 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & -2 & -2 & 0 \\ 0 & 1 & -1 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .\)
  2. \(B\) is not in reduced row echelon form because the pivots are not descending to the right.

Example 40 πŸ”—

  1. Show that

    \[\operatorname{RREF} \left[\begin{array}{ccccc} -1 & -3 & -1 & 0 & -6 \\ -1 & -4 & -2 & 1 & -8 \\ 0 & 2 & 2 & -2 & 4 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & -2 & 3 & 0 \\ 0 & 1 & 1 & -1 & 2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .\]

  2. Explain why the matrix \(B= \left[\begin{array}{ccccc} 1 & 0 & 1 & -2 & 0 \\ 4 & 1 & 5 & -10 & 2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] \) is or is not in reduced row echelon form.

Answer:

  1. \(\operatorname{RREF} \left[\begin{array}{ccccc} -1 & -3 & -1 & 0 & -6 \\ -1 & -4 & -2 & 1 & -8 \\ 0 & 2 & 2 & -2 & 4 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & -2 & 3 & 0 \\ 0 & 1 & 1 & -1 & 2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .\)
  2. \(B\) is not in reduced row echelon form because not every entry above and below each pivot is zero.

Example 41 πŸ”—

  1. Show that

    \[\operatorname{RREF} \left[\begin{array}{ccc} 5 & 1 & 4 \\ 4 & 1 & 3 \\ 2 & -1 & 3 \\ 5 & 1 & 4 \\ 4 & 3 & 1 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & -1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .\]

  2. Explain why the matrix \(B= \left[\begin{array}{ccc} 1 & 0 & -1 \\ 0 & 6 & -6 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \) is or is not in reduced row echelon form.

Answer:

  1. \(\operatorname{RREF} \left[\begin{array}{ccc} 5 & 1 & 4 \\ 4 & 1 & 3 \\ 2 & -1 & 3 \\ 5 & 1 & 4 \\ 4 & 3 & 1 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & -1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .\)
  2. \(B\) is not in reduced row echelon form because the pivots are not all \(1\).

Example 42 πŸ”—

  1. Show that

    \[\operatorname{RREF} \left[\begin{array}{cccc} 1 & -2 & -1 & 2 \\ 3 & -11 & 10 & -2 \\ 0 & -2 & 5 & -3 \\ 3 & -11 & 7 & 1 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -1 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \end{array}\right] .\]

  2. Explain why the matrix \(B= \left[\begin{array}{cccc} 1 & 0 & 4 & -18 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & -4 \\ 0 & 0 & 0 & 0 \end{array}\right] \) is or is not in reduced row echelon form.

Answer:

  1. \(\operatorname{RREF} \left[\begin{array}{cccc} 1 & -2 & -1 & 2 \\ 3 & -11 & 10 & -2 \\ 0 & -2 & 5 & -3 \\ 3 & -11 & 7 & 1 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -1 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \end{array}\right] .\)
  2. \(B\) is not in reduced row echelon form because not every entry above and below each pivot is zero.

Example 43 πŸ”—

  1. Show that

    \[\operatorname{RREF} \left[\begin{array}{ccc} 1 & -4 & 9 \\ -1 & 5 & -11 \\ -2 & 3 & -8 \\ 0 & -3 & 6 \\ 0 & 2 & -4 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & -2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .\]

  2. Explain why the matrix \(B= \left[\begin{array}{ccc} 1 & 0 & 2 \\ -5 & 1 & -11 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \) is or is not in reduced row echelon form.

Answer:

  1. \(\operatorname{RREF} \left[\begin{array}{ccc} 1 & -4 & 9 \\ -1 & 5 & -11 \\ -2 & 3 & -8 \\ 0 & -3 & 6 \\ 0 & 2 & -4 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & -2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .\)
  2. \(B\) is not in reduced row echelon form because not every entry above and below each pivot is zero.

Example 44 πŸ”—

  1. Show that

    \[\operatorname{RREF} \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & -1 \\ 3 & -5 & 5 \\ -4 & 5 & -5 \\ 4 & -3 & 3 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & -1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .\]

  2. Explain why the matrix \(B= \left[\begin{array}{ccc} 0 & 1 & 1 \\ 1 & 0 & -1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \) is or is not in reduced row echelon form.

Answer:

  1. \(\operatorname{RREF} \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & -1 \\ 3 & -5 & 5 \\ -4 & 5 & -5 \\ 4 & -3 & 3 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & -1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .\)
  2. \(B\) is not in reduced row echelon form because the pivots are not descending to the right.

Example 45 πŸ”—

  1. Show that

    \[\operatorname{RREF} \left[\begin{array}{ccc} 1 & 5 & 8 \\ -3 & -8 & -10 \\ -1 & -1 & 0 \\ 0 & 2 & 4 \\ 5 & 10 & 10 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & -2 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .\]

  2. Explain why the matrix \(B= \left[\begin{array}{ccc} 5 & 0 & 5 \\ 0 & 1 & -2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \) is or is not in reduced row echelon form.

Answer:

  1. \(\operatorname{RREF} \left[\begin{array}{ccc} 1 & 5 & 8 \\ -3 & -8 & -10 \\ -1 & -1 & 0 \\ 0 & 2 & 4 \\ 5 & 10 & 10 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & -2 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .\)
  2. \(B\) is not in reduced row echelon form because the pivots are not all \(1\).

Example 46 πŸ”—

  1. Show that

    \[\operatorname{RREF} \left[\begin{array}{cccc} 1 & 3 & 2 & 0 \\ 0 & 1 & -1 & 1 \\ 2 & 10 & 1 & 3 \\ 1 & 2 & -2 & 4 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \end{array}\right] .\]

  2. Explain why the matrix \(B= \left[\begin{array}{cccc} 5 & 0 & 0 & 15 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right] \) is or is not in reduced row echelon form.

Answer:

  1. \(\operatorname{RREF} \left[\begin{array}{cccc} 1 & 3 & 2 & 0 \\ 0 & 1 & -1 & 1 \\ 2 & 10 & 1 & 3 \\ 1 & 2 & -2 & 4 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \end{array}\right] .\)
  2. \(B\) is not in reduced row echelon form because the pivots are not all \(1\).

Example 47 πŸ”—

  1. Show that

    \[\operatorname{RREF} \left[\begin{array}{cccc} 3 & 4 & 2 & -9 \\ 5 & 1 & -8 & 2 \\ 1 & 2 & 2 & -5 \\ 5 & 0 & -10 & 5 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -2 & 1 \\ 0 & 1 & 2 & -3 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] .\]

  2. Explain why the matrix \(B= \left[\begin{array}{cccc} 0 & 1 & -1 & 0 \\ 1 & 0 & 3 & 2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \) is or is not in reduced row echelon form.

Answer:

  1. \(\operatorname{RREF} \left[\begin{array}{cccc} 3 & 4 & 2 & -9 \\ 5 & 1 & -8 & 2 \\ 1 & 2 & 2 & -5 \\ 5 & 0 & -10 & 5 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -2 & 1 \\ 0 & 1 & 2 & -3 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] .\)
  2. \(B\) is not in reduced row echelon form because the pivots are not descending to the right.

Example 48 πŸ”—

  1. Show that

    \[\operatorname{RREF} \left[\begin{array}{ccccc} 1 & -4 & 7 & -4 & 4 \\ 0 & 1 & -2 & 1 & -1 \\ 2 & -5 & 8 & -5 & 5 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & -1 & 0 & 0 \\ 0 & 1 & -2 & 1 & -1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .\]

  2. Explain why the matrix \(B= \left[\begin{array}{ccccc} 0 & 1 & -2 & 2 & 3 \\ 1 & 0 & -1 & -2 & -1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] \) is or is not in reduced row echelon form.

Answer:

  1. \(\operatorname{RREF} \left[\begin{array}{ccccc} 1 & -4 & 7 & -4 & 4 \\ 0 & 1 & -2 & 1 & -1 \\ 2 & -5 & 8 & -5 & 5 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & -1 & 0 & 0 \\ 0 & 1 & -2 & 1 & -1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .\)
  2. \(B\) is not in reduced row echelon form because the pivots are not descending to the right.

Example 49 πŸ”—

  1. Show that

    \[\operatorname{RREF} \left[\begin{array}{ccccc} 1 & 2 & 3 & 5 & -2 \\ -1 & -1 & -2 & -3 & 2 \\ -4 & -4 & -8 & -12 & 8 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 1 & 1 & -2 \\ 0 & 1 & 1 & 2 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .\]

  2. Explain why the matrix \(B= \left[\begin{array}{ccccc} 1 & 0 & 1 & 1 & 1 \\ -2 & 1 & -4 & -4 & -2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] \) is or is not in reduced row echelon form.

Answer:

  1. \(\operatorname{RREF} \left[\begin{array}{ccccc} 1 & 2 & 3 & 5 & -2 \\ -1 & -1 & -2 & -3 & 2 \\ -4 & -4 & -8 & -12 & 8 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 1 & 1 & -2 \\ 0 & 1 & 1 & 2 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .\)
  2. \(B\) is not in reduced row echelon form because not every entry above and below each pivot is zero.

Example 50 πŸ”—

  1. Show that

    \[\operatorname{RREF} \left[\begin{array}{ccccc} 1 & 4 & -4 & -4 & 0 \\ 2 & 9 & -9 & -9 & 0 \\ -3 & -7 & 7 & 7 & 0 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & -1 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .\]

  2. Explain why the matrix \(B= \left[\begin{array}{ccccc} 0 & 0 & 1 & -2 & 0 \\ 1 & -4 & 0 & 0 & -2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] \) is or is not in reduced row echelon form.

Answer:

  1. \(\operatorname{RREF} \left[\begin{array}{ccccc} 1 & 4 & -4 & -4 & 0 \\ 2 & 9 & -9 & -9 & 0 \\ -3 & -7 & 7 & 7 & 0 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & -1 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .\)
  2. \(B\) is not in reduced row echelon form because the pivots are not descending to the right.