Exercise 2

Consider each matrix below and determine if it is in reduced row echelon form. If not, indicate the reason and apply a sequence of row operations to find its reduced row echelon matrix. For each matrix, indicate whether the corresponding linear system has infinitely many solutions, exactly one solution, or no solutions.

1. $$\left[ \begin{array}{rrrr|r} 1 & 1 & 0 & 3 & 3 \\ 0 & 1 & 0 & -2 & 1 \\ 0 & 0 & 1 & 3 & 4 \\ \end{array} \right]$$

2. $$\left[ \begin{array}{rrrr|r} 1 & 0 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 & 0 \\ 0 & 0 & -3 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ \end{array} \right]$$

3. $$\left[ \begin{array}{rrrr|r} 1 & 0 & 0 & 3 & 3 \\ 0 & 1 & 0 & -2 & 1 \\ 0 & 0 & 1 & 3 & 4 \\ 0 & 0 & 0 & 3 & 3 \\ \end{array} \right]$$

4. $$\left[ \begin{array}{rrrr|r} 0 & 0 & 1 & 0 & -1 \\ 0 & 1 & 0 & 0 & 3 \\ 1 & 1 & 1 & 1 & 2 \\ \end{array} \right]$$