Exercise 1

For each of the augmented matrices in reduced row echelon form given below, determine whether the corresponding linear system is consistent and, if so, determine whether the solution is unique. If the system is consistent, identify the free variables and the basic variables and give a description of the solution space in parametric form.

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

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

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

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