1. Show that

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

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

Answer:

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