Activity 1.2.4

Consider each of the following augmented matrices. Determine if the matrix is in reduced row echelon form. If it is not, perform a sequence of scaling, interchange, and replacement operations to obtain a row equivalent matrix that is in reduced row echelon form. Then use the reduced row echelon matrix to describe the solution space.

1. $\left[ \begin{array}{rrr|r} 2 & 0 & 4 & -8 \\ 0 & 1 & 3 & 2 \\ \end{array} \right].$

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

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

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

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