Activity1.2.4Identifying reduced row echelon matrices

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

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

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

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

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

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