1. Show that

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

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

Answer:

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