1. Show that

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

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

Answer:

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