## E2 - Reduced row echelon form

#### Example 1 π

1. Show that

$\operatorname{RREF} \left[\begin{array}{ccccc} 1 & 1 & -2 & -4 & 0 \\ 0 & 1 & -1 & -3 & -3 \\ 0 & -2 & 2 & 6 & 6 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & -1 & -1 & 3 \\ 0 & 1 & -1 & -3 & -3 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .$

2. Explain why the matrix $$B= \left[\begin{array}{ccccc} 1 & 0 & 0 & 1 & -1 \\ 7 & 1 & -1 & 7 & -8 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$$ is or is not in reduced row echelon form.

1. $$\operatorname{RREF} \left[\begin{array}{ccccc} 1 & 1 & -2 & -4 & 0 \\ 0 & 1 & -1 & -3 & -3 \\ 0 & -2 & 2 & 6 & 6 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & -1 & -1 & 3 \\ 0 & 1 & -1 & -3 & -3 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .$$
2. $$B$$ is not in reduced row echelon form because not every entry above and below each pivot is zero.

#### Example 2 π

1. Show that

$\operatorname{RREF} \left[\begin{array}{cccc} 0 & 1 & -1 & 2 \\ -1 & 1 & 4 & -9 \\ 1 & -1 & -3 & 7 \\ 1 & -2 & -4 & 9 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & -2 \\ 0 & 0 & 0 & 0 \end{array}\right] .$

2. Explain why the matrix $$B= \left[\begin{array}{cccc} 1 & 0 & 7 & 22 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 \end{array}\right]$$ is or is not in reduced row echelon form.

1. $$\operatorname{RREF} \left[\begin{array}{cccc} 0 & 1 & -1 & 2 \\ -1 & 1 & 4 & -9 \\ 1 & -1 & -3 & 7 \\ 1 & -2 & -4 & 9 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & -2 \\ 0 & 0 & 0 & 0 \end{array}\right] .$$
2. $$B$$ is not in reduced row echelon form because not every entry above and below each pivot is zero.

#### Example 3 π

1. Show that

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & 10 & -10 & -8 \\ 1 & 2 & 1 & 9 \\ 0 & -3 & 4 & 6 \\ -1 & -6 & 1 & -11 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 \end{array}\right] .$

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

1. $$\operatorname{RREF} \left[\begin{array}{cccc} 1 & 10 & -10 & -8 \\ 1 & 2 & 1 & 9 \\ 0 & -3 & 4 & 6 \\ -1 & -6 & 1 & -11 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 \end{array}\right] .$$
2. $$B$$ is not in reduced row echelon form because the pivots are not all $$1$$.

#### Example 4 π

1. Show that

$\operatorname{RREF} \left[\begin{array}{ccccc} 3 & 12 & 12 & 1 & 8 \\ -2 & -8 & -8 & 7 & 10 \\ 1 & 4 & 4 & 2 & 6 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 4 & 4 & 0 & 2 \\ 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .$

2. Explain why the matrix $$B= \left[\begin{array}{ccccc} 0 & 1 & 2 & 0 & 0 \\ 1 & 0 & 2 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$$ is or is not in reduced row echelon form.

1. $$\operatorname{RREF} \left[\begin{array}{ccccc} 3 & 12 & 12 & 1 & 8 \\ -2 & -8 & -8 & 7 & 10 \\ 1 & 4 & 4 & 2 & 6 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 4 & 4 & 0 & 2 \\ 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .$$
2. $$B$$ is not in reduced row echelon form because the pivots are not descending to the right.

#### Example 5 π

1. Show that

$\operatorname{RREF} \left[\begin{array}{ccccc} 1 & 4 & -2 & -5 & 4 \\ 0 & 0 & 1 & 2 & -2 \\ 0 & 0 & -3 & -6 & 6 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 4 & 0 & -1 & 0 \\ 0 & 0 & 1 & 2 & -2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .$

2. Explain why the matrix $$B= \left[\begin{array}{ccccc} 4 & 0 & -4 & 0 & -4 \\ 0 & 1 & 1 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$$ is or is not in reduced row echelon form.

1. $$\operatorname{RREF} \left[\begin{array}{ccccc} 1 & 4 & -2 & -5 & 4 \\ 0 & 0 & 1 & 2 & -2 \\ 0 & 0 & -3 & -6 & 6 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 4 & 0 & -1 & 0 \\ 0 & 0 & 1 & 2 & -2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .$$
2. $$B$$ is not in reduced row echelon form because the pivots are not all $$1$$.

#### Example 6 π

1. Show that

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & -1 & -9 & 6 \\ 0 & 1 & 4 & -3 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 4 & -6 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -5 & 0 \\ 0 & 1 & 4 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right] .$

2. Explain why the matrix $$B= \left[\begin{array}{cccc} 1 & 0 & 0 & -2 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -4 & -4 \\ 0 & 0 & 0 & 0 \end{array}\right]$$ is or is not in reduced row echelon form.

1. $$\operatorname{RREF} \left[\begin{array}{cccc} 1 & -1 & -9 & 6 \\ 0 & 1 & 4 & -3 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 4 & -6 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -5 & 0 \\ 0 & 1 & 4 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right] .$$
2. $$B$$ is not in reduced row echelon form because the pivots are not all $$1$$.

#### Example 7 π

1. Show that

$\operatorname{RREF} \left[\begin{array}{ccc} -8 & 4 & 4 \\ -1 & 3 & 8 \\ -3 & 2 & 3 \\ 1 & -2 & -5 \\ -2 & 4 & 10 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & 3 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .$

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

1. $$\operatorname{RREF} \left[\begin{array}{ccc} -8 & 4 & 4 \\ -1 & 3 & 8 \\ -3 & 2 & 3 \\ 1 & -2 & -5 \\ -2 & 4 & 10 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & 3 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .$$
2. $$B$$ is not in reduced row echelon form because not every entry above and below each pivot is zero.

#### Example 8 π

1. Show that

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

2. Explain why the matrix $$B= \left[\begin{array}{ccc} 1 & 4 & -4 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]$$ is or is not in reduced row echelon form.

1. $$\operatorname{RREF} \left[\begin{array}{ccc} -2 & -2 & -2 \\ 0 & 1 & 2 \\ 0 & -2 & -4 \\ 0 & 4 & 8 \\ 1 & -1 & -3 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & -1 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .$$
2. $$B$$ is not in reduced row echelon form because not every entry above and below each pivot is zero.

#### Example 9 π

1. Show that

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & 0 & 1 & 1 \\ 0 & 1 & 5 & -11 \\ -4 & -1 & -8 & 5 \\ 5 & 1 & 6 & 2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 3 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & -2 \\ 0 & 0 & 0 & 0 \end{array}\right] .$

2. Explain why the matrix $$B= \left[\begin{array}{cccc} 0 & 0 & 1 & 2 \\ 0 & 1 & 0 & 2 \\ 1 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 \end{array}\right]$$ is or is not in reduced row echelon form.

1. $$\operatorname{RREF} \left[\begin{array}{cccc} 1 & 0 & 1 & 1 \\ 0 & 1 & 5 & -11 \\ -4 & -1 & -8 & 5 \\ 5 & 1 & 6 & 2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 3 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & -2 \\ 0 & 0 & 0 & 0 \end{array}\right] .$$
2. $$B$$ is not in reduced row echelon form because the pivots are not descending to the right.

#### Example 10 π

1. Show that

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & 3 & 10 & 6 \\ 0 & 1 & 3 & 2 \\ 0 & -1 & -3 & -2 \\ -1 & -2 & -7 & -4 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 1 & 0 \\ 0 & 1 & 3 & 2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] .$

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

1. $$\operatorname{RREF} \left[\begin{array}{cccc} 1 & 3 & 10 & 6 \\ 0 & 1 & 3 & 2 \\ 0 & -1 & -3 & -2 \\ -1 & -2 & -7 & -4 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 1 & 0 \\ 0 & 1 & 3 & 2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] .$$
2. $$B$$ is not in reduced row echelon form because not every entry above and below each pivot is zero.

#### Example 11 π

1. Show that

$\operatorname{RREF} \left[\begin{array}{ccccc} 1 & -1 & 2 & 3 & 2 \\ -2 & 3 & -4 & -8 & -6 \\ 5 & 0 & 10 & 5 & 0 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 2 & 1 & 0 \\ 0 & 1 & 0 & -2 & -2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .$

2. Explain why the matrix $$B= \left[\begin{array}{ccccc} 1 & 5 & 0 & -3 & 2 \\ 0 & 0 & -6 & -12 & 6 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$$ is or is not in reduced row echelon form.

1. $$\operatorname{RREF} \left[\begin{array}{ccccc} 1 & -1 & 2 & 3 & 2 \\ -2 & 3 & -4 & -8 & -6 \\ 5 & 0 & 10 & 5 & 0 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 2 & 1 & 0 \\ 0 & 1 & 0 & -2 & -2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .$$
2. $$B$$ is not in reduced row echelon form because the pivots are not all $$1$$.

#### Example 12 π

1. Show that

$\operatorname{RREF} \left[\begin{array}{ccc} 1 & 0 & 3 \\ -1 & 1 & -6 \\ 3 & 4 & -3 \\ -1 & -4 & 9 \\ -1 & -1 & 0 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 3 \\ 0 & 1 & -3 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .$

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

1. $$\operatorname{RREF} \left[\begin{array}{ccc} 1 & 0 & 3 \\ -1 & 1 & -6 \\ 3 & 4 & -3 \\ -1 & -4 & 9 \\ -1 & -1 & 0 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 3 \\ 0 & 1 & -3 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .$$
2. $$B$$ is not in reduced row echelon form because the pivots are not descending to the right.

#### Example 13 π

1. Show that

$\operatorname{RREF} \left[\begin{array}{ccc} -2 & 3 & 6 \\ 3 & 10 & -9 \\ -4 & -11 & 12 \\ -1 & 0 & 3 \\ -3 & -10 & 9 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & -3 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .$

2. Explain why the matrix $$B= \left[\begin{array}{ccc} 1 & 0 & -1 \\ 0 & -4 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]$$ is or is not in reduced row echelon form.

1. $$\operatorname{RREF} \left[\begin{array}{ccc} -2 & 3 & 6 \\ 3 & 10 & -9 \\ -4 & -11 & 12 \\ -1 & 0 & 3 \\ -3 & -10 & 9 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & -3 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .$$
2. $$B$$ is not in reduced row echelon form because the pivots are not all $$1$$.

#### Example 14 π

1. Show that

$\operatorname{RREF} \left[\begin{array}{ccc} -2 & 8 & -6 \\ -1 & 5 & -4 \\ 3 & -7 & 4 \\ 0 & 4 & -4 \\ 1 & -4 & 3 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & -1 \\ 0 & 1 & -1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .$

2. Explain why the matrix $$B= \left[\begin{array}{ccc} -6 & 0 & -6 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]$$ is or is not in reduced row echelon form.

1. $$\operatorname{RREF} \left[\begin{array}{ccc} -2 & 8 & -6 \\ -1 & 5 & -4 \\ 3 & -7 & 4 \\ 0 & 4 & -4 \\ 1 & -4 & 3 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & -1 \\ 0 & 1 & -1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .$$
2. $$B$$ is not in reduced row echelon form because the pivots are not all $$1$$.

#### Example 15 π

1. Show that

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

2. Explain why the matrix $$B= \left[\begin{array}{ccccc} 1 & -4 & 2 & 9 & -9 \\ 0 & 1 & 0 & -2 & 2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$$ is or is not in reduced row echelon form.

1. $$\operatorname{RREF} \left[\begin{array}{ccccc} 1 & -5 & 3 & -2 & 4 \\ 0 & 0 & 1 & -1 & 1 \\ 0 & 0 & 5 & -5 & 5 \end{array}\right] = \left[\begin{array}{ccccc} 1 & -5 & 0 & 1 & 1 \\ 0 & 0 & 1 & -1 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .$$
2. $$B$$ is not in reduced row echelon form because not every entry above and below each pivot is zero.

#### Example 16 π

1. Show that

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & 6 & 11 & 10 \\ -2 & -12 & -3 & -1 \\ 0 & 0 & -3 & -3 \\ -2 & -12 & -8 & -6 \end{array}\right] = \left[\begin{array}{cccc} 1 & 6 & 0 & -1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] .$

2. Explain why the matrix $$B= \left[\begin{array}{cccc} 0 & 0 & 1 & 1 \\ 1 & -4 & 0 & -2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$$ is or is not in reduced row echelon form.

1. $$\operatorname{RREF} \left[\begin{array}{cccc} 1 & 6 & 11 & 10 \\ -2 & -12 & -3 & -1 \\ 0 & 0 & -3 & -3 \\ -2 & -12 & -8 & -6 \end{array}\right] = \left[\begin{array}{cccc} 1 & 6 & 0 & -1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] .$$
2. $$B$$ is not in reduced row echelon form because the pivots are not descending to the right.

#### Example 17 π

1. Show that

$\operatorname{RREF} \left[\begin{array}{ccc} -1 & -2 & -5 \\ 0 & 1 & 2 \\ -3 & 5 & 7 \\ -2 & 4 & 6 \\ -2 & 1 & 0 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .$

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

1. $$\operatorname{RREF} \left[\begin{array}{ccc} -1 & -2 & -5 \\ 0 & 1 & 2 \\ -3 & 5 & 7 \\ -2 & 4 & 6 \\ -2 & 1 & 0 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .$$
2. $$B$$ is not in reduced row echelon form because the pivots are not descending to the right.

#### Example 18 π

1. Show that

$\operatorname{RREF} \left[\begin{array}{ccc} 0 & -1 & -2 \\ 1 & -1 & -4 \\ 1 & -1 & -4 \\ 1 & -2 & -6 \\ -2 & 4 & 12 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & -2 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .$

2. Explain why the matrix $$B= \left[\begin{array}{ccc} 1 & 4 & 0 \\ 0 & 0 & -3 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]$$ is or is not in reduced row echelon form.

1. $$\operatorname{RREF} \left[\begin{array}{ccc} 0 & -1 & -2 \\ 1 & -1 & -4 \\ 1 & -1 & -4 \\ 1 & -2 & -6 \\ -2 & 4 & 12 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & -2 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .$$
2. $$B$$ is not in reduced row echelon form because the pivots are not all $$1$$.

#### Example 19 π

1. Show that

$\operatorname{RREF} \left[\begin{array}{ccccc} 1 & 1 & 1 & 0 & 4 \\ 0 & 1 & 2 & 2 & 2 \\ 0 & 1 & 2 & 2 & 2 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & -1 & -2 & 2 \\ 0 & 1 & 2 & 2 & 2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .$

2. Explain why the matrix $$B= \left[\begin{array}{ccccc} 1 & -5 & 0 & -3 & -3 \\ 0 & 0 & -2 & 2 & -2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$$ is or is not in reduced row echelon form.

1. $$\operatorname{RREF} \left[\begin{array}{ccccc} 1 & 1 & 1 & 0 & 4 \\ 0 & 1 & 2 & 2 & 2 \\ 0 & 1 & 2 & 2 & 2 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & -1 & -2 & 2 \\ 0 & 1 & 2 & 2 & 2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .$$
2. $$B$$ is not in reduced row echelon form because the pivots are not all $$1$$.

#### Example 20 π

1. Show that

$\operatorname{RREF} \left[\begin{array}{ccc} -2 & 5 & 7 \\ -2 & -9 & -7 \\ -1 & -3 & -2 \\ -1 & 0 & 1 \\ 0 & -3 & -3 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & -1 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .$

2. Explain why the matrix $$B= \left[\begin{array}{ccc} 1 & 0 & 2 \\ -5 & 1 & -8 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]$$ is or is not in reduced row echelon form.

1. $$\operatorname{RREF} \left[\begin{array}{ccc} -2 & 5 & 7 \\ -2 & -9 & -7 \\ -1 & -3 & -2 \\ -1 & 0 & 1 \\ 0 & -3 & -3 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & -1 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .$$
2. $$B$$ is not in reduced row echelon form because not every entry above and below each pivot is zero.

#### Example 21 π

1. Show that

$\operatorname{RREF} \left[\begin{array}{ccccc} 0 & -1 & 0 & 1 & -1 \\ 1 & 4 & 3 & -5 & 4 \\ 2 & 7 & 6 & -9 & 7 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 3 & -1 & 0 \\ 0 & 1 & 0 & -1 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .$

2. Explain why the matrix $$B= \left[\begin{array}{ccccc} 7 & 0 & 21 & -14 & 14 \\ 0 & 1 & 1 & -1 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$$ is or is not in reduced row echelon form.

1. $$\operatorname{RREF} \left[\begin{array}{ccccc} 0 & -1 & 0 & 1 & -1 \\ 1 & 4 & 3 & -5 & 4 \\ 2 & 7 & 6 & -9 & 7 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 3 & -1 & 0 \\ 0 & 1 & 0 & -1 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .$$
2. $$B$$ is not in reduced row echelon form because the pivots are not all $$1$$.

#### Example 22 π

1. Show that

$\operatorname{RREF} \left[\begin{array}{ccccc} 1 & -4 & 5 & -6 & 10 \\ 0 & 0 & 0 & 1 & -2 \\ 2 & -8 & 10 & -7 & 10 \end{array}\right] = \left[\begin{array}{ccccc} 1 & -4 & 5 & 0 & -2 \\ 0 & 0 & 0 & 1 & -2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .$

2. Explain why the matrix $$B= \left[\begin{array}{ccccc} 1 & 0 & -1 & -2 & 1 \\ -7 & 1 & 7 & 14 & -8 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$$ is or is not in reduced row echelon form.

1. $$\operatorname{RREF} \left[\begin{array}{ccccc} 1 & -4 & 5 & -6 & 10 \\ 0 & 0 & 0 & 1 & -2 \\ 2 & -8 & 10 & -7 & 10 \end{array}\right] = \left[\begin{array}{ccccc} 1 & -4 & 5 & 0 & -2 \\ 0 & 0 & 0 & 1 & -2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .$$
2. $$B$$ is not in reduced row echelon form because not every entry above and below each pivot is zero.

#### Example 23 π

1. Show that

$\operatorname{RREF} \left[\begin{array}{ccc} 1 & -5 & 12 \\ 0 & 1 & -2 \\ -1 & 3 & -8 \\ 0 & 1 & -2 \\ 1 & -4 & 10 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 2 \\ 0 & 1 & -2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .$

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

1. $$\operatorname{RREF} \left[\begin{array}{ccc} 1 & -5 & 12 \\ 0 & 1 & -2 \\ -1 & 3 & -8 \\ 0 & 1 & -2 \\ 1 & -4 & 10 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 2 \\ 0 & 1 & -2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .$$
2. $$B$$ is not in reduced row echelon form because the pivots are not descending to the right.

#### Example 24 π

1. Show that

$\operatorname{RREF} \left[\begin{array}{ccccc} -4 & 1 & 6 & 8 & 3 \\ -3 & 4 & 11 & 6 & -1 \\ -5 & 2 & 9 & 10 & 3 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & -1 & -2 & -1 \\ 0 & 1 & 2 & 0 & -1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .$

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

1. $$\operatorname{RREF} \left[\begin{array}{ccccc} -4 & 1 & 6 & 8 & 3 \\ -3 & 4 & 11 & 6 & -1 \\ -5 & 2 & 9 & 10 & 3 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & -1 & -2 & -1 \\ 0 & 1 & 2 & 0 & -1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .$$
2. $$B$$ is not in reduced row echelon form because the pivots are not descending to the right.

#### Example 25 π

1. Show that

$\operatorname{RREF} \left[\begin{array}{cccc} -2 & 10 & -3 & -9 \\ -1 & 5 & 4 & 1 \\ -1 & 5 & -1 & -4 \\ 1 & -5 & 0 & 3 \end{array}\right] = \left[\begin{array}{cccc} 1 & -5 & 0 & 3 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] .$

2. Explain why the matrix $$B= \left[\begin{array}{cccc} 0 & 1 & -2 & 4 \\ 1 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$$ is or is not in reduced row echelon form.

1. $$\operatorname{RREF} \left[\begin{array}{cccc} -2 & 10 & -3 & -9 \\ -1 & 5 & 4 & 1 \\ -1 & 5 & -1 & -4 \\ 1 & -5 & 0 & 3 \end{array}\right] = \left[\begin{array}{cccc} 1 & -5 & 0 & 3 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] .$$
2. $$B$$ is not in reduced row echelon form because the pivots are not descending to the right.

#### Example 26 π

1. Show that

$\operatorname{RREF} \left[\begin{array}{ccc} 1 & -7 & -7 \\ 2 & 11 & 11 \\ 3 & 11 & 11 \\ 0 & -4 & -4 \\ 2 & 7 & 7 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .$

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

1. $$\operatorname{RREF} \left[\begin{array}{ccc} 1 & -7 & -7 \\ 2 & 11 & 11 \\ 3 & 11 & 11 \\ 0 & -4 & -4 \\ 2 & 7 & 7 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .$$
2. $$B$$ is not in reduced row echelon form because the pivots are not descending to the right.

#### Example 27 π

1. Show that

$\operatorname{RREF} \left[\begin{array}{ccccc} -9 & 3 & -3 & -12 & 0 \\ -2 & 3 & -3 & 2 & 7 \\ 5 & -2 & 2 & 6 & -1 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 0 & 2 & 1 \\ 0 & 1 & -1 & 2 & 3 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .$

2. Explain why the matrix $$B= \left[\begin{array}{ccccc} 1 & 0 & 1 & 0 & 1 \\ 0 & -4 & 0 & -8 & 4 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$$ is or is not in reduced row echelon form.

1. $$\operatorname{RREF} \left[\begin{array}{ccccc} -9 & 3 & -3 & -12 & 0 \\ -2 & 3 & -3 & 2 & 7 \\ 5 & -2 & 2 & 6 & -1 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 0 & 2 & 1 \\ 0 & 1 & -1 & 2 & 3 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .$$
2. $$B$$ is not in reduced row echelon form because the pivots are not all $$1$$.

#### Example 28 π

1. Show that

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & 1 & 2 & -7 \\ 0 & 1 & 4 & -9 \\ 0 & -1 & -3 & 7 \\ 2 & 0 & -7 & 10 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -2 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & -2 \\ 0 & 0 & 0 & 0 \end{array}\right] .$

2. Explain why the matrix $$B= \left[\begin{array}{cccc} 1 & 0 & 7 & 22 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 \end{array}\right]$$ is or is not in reduced row echelon form.

1. $$\operatorname{RREF} \left[\begin{array}{cccc} 1 & 1 & 2 & -7 \\ 0 & 1 & 4 & -9 \\ 0 & -1 & -3 & 7 \\ 2 & 0 & -7 & 10 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -2 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & -2 \\ 0 & 0 & 0 & 0 \end{array}\right] .$$
2. $$B$$ is not in reduced row echelon form because not every entry above and below each pivot is zero.

#### Example 29 π

1. Show that

$\operatorname{RREF} \left[\begin{array}{ccc} 1 & -2 & 2 \\ -4 & 9 & -9 \\ -4 & 8 & -8 \\ -5 & 10 & -10 \\ -5 & 12 & -12 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & -1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .$

2. Explain why the matrix $$B= \left[\begin{array}{ccc} 1 & 0 & 0 \\ -4 & 1 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]$$ is or is not in reduced row echelon form.

1. $$\operatorname{RREF} \left[\begin{array}{ccc} 1 & -2 & 2 \\ -4 & 9 & -9 \\ -4 & 8 & -8 \\ -5 & 10 & -10 \\ -5 & 12 & -12 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & -1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .$$
2. $$B$$ is not in reduced row echelon form because not every entry above and below each pivot is zero.

#### Example 30 π

1. Show that

$\operatorname{RREF} \left[\begin{array}{ccccc} 1 & -4 & 1 & -3 & 4 \\ -1 & 4 & 0 & 0 & -1 \\ 1 & -4 & 1 & -3 & 4 \end{array}\right] = \left[\begin{array}{ccccc} 1 & -4 & 0 & 0 & 1 \\ 0 & 0 & 1 & -3 & 3 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .$

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

1. $$\operatorname{RREF} \left[\begin{array}{ccccc} 1 & -4 & 1 & -3 & 4 \\ -1 & 4 & 0 & 0 & -1 \\ 1 & -4 & 1 & -3 & 4 \end{array}\right] = \left[\begin{array}{ccccc} 1 & -4 & 0 & 0 & 1 \\ 0 & 0 & 1 & -3 & 3 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .$$
2. $$B$$ is not in reduced row echelon form because not every entry above and below each pivot is zero.

#### Example 31 π

1. Show that

$\operatorname{RREF} \left[\begin{array}{cccc} 0 & 5 & 11 & 12 \\ -2 & -5 & -6 & 2 \\ -1 & 2 & 7 & 12 \\ -2 & -4 & -3 & 6 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -2 \\ 0 & 1 & 0 & -2 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 \end{array}\right] .$

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

1. $$\operatorname{RREF} \left[\begin{array}{cccc} 0 & 5 & 11 & 12 \\ -2 & -5 & -6 & 2 \\ -1 & 2 & 7 & 12 \\ -2 & -4 & -3 & 6 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -2 \\ 0 & 1 & 0 & -2 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 \end{array}\right] .$$
2. $$B$$ is not in reduced row echelon form because the pivots are not all $$1$$.

#### Example 32 π

1. Show that

$\operatorname{RREF} \left[\begin{array}{ccccc} 1 & -5 & 12 & 7 & 12 \\ 0 & 1 & -2 & -1 & -2 \\ -1 & 4 & -10 & -6 & -10 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 2 & 2 & 2 \\ 0 & 1 & -2 & -1 & -2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .$

2. Explain why the matrix $$B= \left[\begin{array}{ccccc} 1 & 0 & 1 & 1 & -1 \\ -3 & 1 & -5 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$$ is or is not in reduced row echelon form.

1. $$\operatorname{RREF} \left[\begin{array}{ccccc} 1 & -5 & 12 & 7 & 12 \\ 0 & 1 & -2 & -1 & -2 \\ -1 & 4 & -10 & -6 & -10 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 2 & 2 & 2 \\ 0 & 1 & -2 & -1 & -2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .$$
2. $$B$$ is not in reduced row echelon form because not every entry above and below each pivot is zero.

#### Example 33 π

1. Show that

$\operatorname{RREF} \left[\begin{array}{ccc} 1 & -1 & 3 \\ 2 & -1 & 3 \\ 3 & -2 & 6 \\ 4 & -1 & 3 \\ -4 & 2 & -6 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & -3 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .$

2. Explain why the matrix $$B= \left[\begin{array}{ccc} 1 & 3 & -4 \\ 0 & 1 & -2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]$$ is or is not in reduced row echelon form.

1. $$\operatorname{RREF} \left[\begin{array}{ccc} 1 & -1 & 3 \\ 2 & -1 & 3 \\ 3 & -2 & 6 \\ 4 & -1 & 3 \\ -4 & 2 & -6 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & -3 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .$$
2. $$B$$ is not in reduced row echelon form because not every entry above and below each pivot is zero.

#### Example 34 π

1. Show that

$\operatorname{RREF} \left[\begin{array}{ccccc} 1 & 4 & 5 & -7 & 6 \\ -1 & -3 & -3 & 5 & -5 \\ -2 & -4 & -2 & 6 & -8 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & -3 & 1 & 2 \\ 0 & 1 & 2 & -2 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .$

2. Explain why the matrix $$B= \left[\begin{array}{ccccc} 1 & 3 & 1 & -11 & 4 \\ 0 & 1 & 0 & -3 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$$ is or is not in reduced row echelon form.

1. $$\operatorname{RREF} \left[\begin{array}{ccccc} 1 & 4 & 5 & -7 & 6 \\ -1 & -3 & -3 & 5 & -5 \\ -2 & -4 & -2 & 6 & -8 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & -3 & 1 & 2 \\ 0 & 1 & 2 & -2 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .$$
2. $$B$$ is not in reduced row echelon form because not every entry above and below each pivot is zero.

#### Example 35 π

1. Show that

$\operatorname{RREF} \left[\begin{array}{cccc} -2 & -4 & 12 & -12 \\ -1 & -4 & 10 & -10 \\ -1 & -3 & 8 & -8 \\ 0 & -2 & 4 & -4 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -2 & 2 \\ 0 & 1 & -2 & 2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] .$

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

1. $$\operatorname{RREF} \left[\begin{array}{cccc} -2 & -4 & 12 & -12 \\ -1 & -4 & 10 & -10 \\ -1 & -3 & 8 & -8 \\ 0 & -2 & 4 & -4 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -2 & 2 \\ 0 & 1 & -2 & 2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] .$$
2. $$B$$ is not in reduced row echelon form because the pivots are not descending to the right.

#### Example 36 π

1. Show that

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & -1 & -1 & -1 \\ 2 & -1 & 1 & 1 \\ -4 & 2 & -1 & -1 \\ 0 & -3 & -10 & -10 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right] .$

2. Explain why the matrix $$B= \left[\begin{array}{cccc} 1 & 0 & 0 & -2 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & -6 & -12 \\ 0 & 0 & 0 & 0 \end{array}\right]$$ is or is not in reduced row echelon form.

1. $$\operatorname{RREF} \left[\begin{array}{cccc} 1 & -1 & -1 & -1 \\ 2 & -1 & 1 & 1 \\ -4 & 2 & -1 & -1 \\ 0 & -3 & -10 & -10 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right] .$$
2. $$B$$ is not in reduced row echelon form because the pivots are not all $$1$$.

#### Example 37 π

1. Show that

$\operatorname{RREF} \left[\begin{array}{cccc} -3 & 3 & 6 & 3 \\ -1 & 1 & 2 & 1 \\ 2 & -3 & -6 & 0 \\ -4 & 3 & 6 & 6 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -3 \\ 0 & 1 & 2 & -2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] .$

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

1. $$\operatorname{RREF} \left[\begin{array}{cccc} -3 & 3 & 6 & 3 \\ -1 & 1 & 2 & 1 \\ 2 & -3 & -6 & 0 \\ -4 & 3 & 6 & 6 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -3 \\ 0 & 1 & 2 & -2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] .$$
2. $$B$$ is not in reduced row echelon form because the pivots are not descending to the right.

#### Example 38 π

1. Show that

$\operatorname{RREF} \left[\begin{array}{ccc} -1 & -7 & 11 \\ 1 & 6 & -9 \\ -1 & -1 & -1 \\ 1 & 4 & -5 \\ 3 & 10 & -11 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 3 \\ 0 & 1 & -2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .$

2. Explain why the matrix $$B= \left[\begin{array}{ccc} 1 & -5 & 15 \\ 0 & 1 & -3 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]$$ is or is not in reduced row echelon form.

1. $$\operatorname{RREF} \left[\begin{array}{ccc} -1 & -7 & 11 \\ 1 & 6 & -9 \\ -1 & -1 & -1 \\ 1 & 4 & -5 \\ 3 & 10 & -11 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 3 \\ 0 & 1 & -2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .$$
2. $$B$$ is not in reduced row echelon form because not every entry above and below each pivot is zero.

#### Example 39 π

1. Show that

$\operatorname{RREF} \left[\begin{array}{ccccc} 0 & -1 & 1 & -1 & -2 \\ 1 & -3 & 1 & -5 & -6 \\ 0 & 3 & -3 & 3 & 6 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & -2 & -2 & 0 \\ 0 & 1 & -1 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .$

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

1. $$\operatorname{RREF} \left[\begin{array}{ccccc} 0 & -1 & 1 & -1 & -2 \\ 1 & -3 & 1 & -5 & -6 \\ 0 & 3 & -3 & 3 & 6 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & -2 & -2 & 0 \\ 0 & 1 & -1 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .$$
2. $$B$$ is not in reduced row echelon form because the pivots are not descending to the right.

#### Example 40 π

1. Show that

$\operatorname{RREF} \left[\begin{array}{ccccc} -1 & -3 & -1 & 0 & -6 \\ -1 & -4 & -2 & 1 & -8 \\ 0 & 2 & 2 & -2 & 4 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & -2 & 3 & 0 \\ 0 & 1 & 1 & -1 & 2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .$

2. Explain why the matrix $$B= \left[\begin{array}{ccccc} 1 & 0 & 1 & -2 & 0 \\ 4 & 1 & 5 & -10 & 2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$$ is or is not in reduced row echelon form.

1. $$\operatorname{RREF} \left[\begin{array}{ccccc} -1 & -3 & -1 & 0 & -6 \\ -1 & -4 & -2 & 1 & -8 \\ 0 & 2 & 2 & -2 & 4 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & -2 & 3 & 0 \\ 0 & 1 & 1 & -1 & 2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .$$
2. $$B$$ is not in reduced row echelon form because not every entry above and below each pivot is zero.

#### Example 41 π

1. Show that

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

2. Explain why the matrix $$B= \left[\begin{array}{ccc} 1 & 0 & -1 \\ 0 & 6 & -6 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]$$ is or is not in reduced row echelon form.

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

#### Example 42 π

1. Show that

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & -2 & -1 & 2 \\ 3 & -11 & 10 & -2 \\ 0 & -2 & 5 & -3 \\ 3 & -11 & 7 & 1 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -1 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \end{array}\right] .$

2. Explain why the matrix $$B= \left[\begin{array}{cccc} 1 & 0 & 4 & -18 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & -4 \\ 0 & 0 & 0 & 0 \end{array}\right]$$ is or is not in reduced row echelon form.

1. $$\operatorname{RREF} \left[\begin{array}{cccc} 1 & -2 & -1 & 2 \\ 3 & -11 & 10 & -2 \\ 0 & -2 & 5 & -3 \\ 3 & -11 & 7 & 1 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -1 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \end{array}\right] .$$
2. $$B$$ is not in reduced row echelon form because not every entry above and below each pivot is zero.

#### Example 43 π

1. Show that

$\operatorname{RREF} \left[\begin{array}{ccc} 1 & -4 & 9 \\ -1 & 5 & -11 \\ -2 & 3 & -8 \\ 0 & -3 & 6 \\ 0 & 2 & -4 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & -2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .$

2. Explain why the matrix $$B= \left[\begin{array}{ccc} 1 & 0 & 2 \\ -5 & 1 & -11 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]$$ is or is not in reduced row echelon form.

1. $$\operatorname{RREF} \left[\begin{array}{ccc} 1 & -4 & 9 \\ -1 & 5 & -11 \\ -2 & 3 & -8 \\ 0 & -3 & 6 \\ 0 & 2 & -4 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & -2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .$$
2. $$B$$ is not in reduced row echelon form because not every entry above and below each pivot is zero.

#### Example 44 π

1. Show that

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

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

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

#### Example 45 π

1. Show that

$\operatorname{RREF} \left[\begin{array}{ccc} 1 & 5 & 8 \\ -3 & -8 & -10 \\ -1 & -1 & 0 \\ 0 & 2 & 4 \\ 5 & 10 & 10 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & -2 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .$

2. Explain why the matrix $$B= \left[\begin{array}{ccc} 5 & 0 & 5 \\ 0 & 1 & -2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]$$ is or is not in reduced row echelon form.

1. $$\operatorname{RREF} \left[\begin{array}{ccc} 1 & 5 & 8 \\ -3 & -8 & -10 \\ -1 & -1 & 0 \\ 0 & 2 & 4 \\ 5 & 10 & 10 \end{array}\right] = \left[\begin{array}{ccc} 1 & 0 & -2 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] .$$
2. $$B$$ is not in reduced row echelon form because the pivots are not all $$1$$.

#### Example 46 π

1. Show that

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & 3 & 2 & 0 \\ 0 & 1 & -1 & 1 \\ 2 & 10 & 1 & 3 \\ 1 & 2 & -2 & 4 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \end{array}\right] .$

2. Explain why the matrix $$B= \left[\begin{array}{cccc} 5 & 0 & 0 & 15 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right]$$ is or is not in reduced row echelon form.

1. $$\operatorname{RREF} \left[\begin{array}{cccc} 1 & 3 & 2 & 0 \\ 0 & 1 & -1 & 1 \\ 2 & 10 & 1 & 3 \\ 1 & 2 & -2 & 4 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \end{array}\right] .$$
2. $$B$$ is not in reduced row echelon form because the pivots are not all $$1$$.

#### Example 47 π

1. Show that

$\operatorname{RREF} \left[\begin{array}{cccc} 3 & 4 & 2 & -9 \\ 5 & 1 & -8 & 2 \\ 1 & 2 & 2 & -5 \\ 5 & 0 & -10 & 5 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -2 & 1 \\ 0 & 1 & 2 & -3 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] .$

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

1. $$\operatorname{RREF} \left[\begin{array}{cccc} 3 & 4 & 2 & -9 \\ 5 & 1 & -8 & 2 \\ 1 & 2 & 2 & -5 \\ 5 & 0 & -10 & 5 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -2 & 1 \\ 0 & 1 & 2 & -3 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] .$$
2. $$B$$ is not in reduced row echelon form because the pivots are not descending to the right.

#### Example 48 π

1. Show that

$\operatorname{RREF} \left[\begin{array}{ccccc} 1 & -4 & 7 & -4 & 4 \\ 0 & 1 & -2 & 1 & -1 \\ 2 & -5 & 8 & -5 & 5 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & -1 & 0 & 0 \\ 0 & 1 & -2 & 1 & -1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .$

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

1. $$\operatorname{RREF} \left[\begin{array}{ccccc} 1 & -4 & 7 & -4 & 4 \\ 0 & 1 & -2 & 1 & -1 \\ 2 & -5 & 8 & -5 & 5 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & -1 & 0 & 0 \\ 0 & 1 & -2 & 1 & -1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .$$
2. $$B$$ is not in reduced row echelon form because the pivots are not descending to the right.

#### Example 49 π

1. Show that

$\operatorname{RREF} \left[\begin{array}{ccccc} 1 & 2 & 3 & 5 & -2 \\ -1 & -1 & -2 & -3 & 2 \\ -4 & -4 & -8 & -12 & 8 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 1 & 1 & -2 \\ 0 & 1 & 1 & 2 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .$

2. Explain why the matrix $$B= \left[\begin{array}{ccccc} 1 & 0 & 1 & 1 & 1 \\ -2 & 1 & -4 & -4 & -2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$$ is or is not in reduced row echelon form.

1. $$\operatorname{RREF} \left[\begin{array}{ccccc} 1 & 2 & 3 & 5 & -2 \\ -1 & -1 & -2 & -3 & 2 \\ -4 & -4 & -8 & -12 & 8 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 1 & 1 & -2 \\ 0 & 1 & 1 & 2 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .$$
2. $$B$$ is not in reduced row echelon form because not every entry above and below each pivot is zero.

#### Example 50 π

1. Show that

$\operatorname{RREF} \left[\begin{array}{ccccc} 1 & 4 & -4 & -4 & 0 \\ 2 & 9 & -9 & -9 & 0 \\ -3 & -7 & 7 & 7 & 0 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & -1 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .$

2. Explain why the matrix $$B= \left[\begin{array}{ccccc} 0 & 0 & 1 & -2 & 0 \\ 1 & -4 & 0 & 0 & -2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$$ is or is not in reduced row echelon form.

1. $$\operatorname{RREF} \left[\begin{array}{ccccc} 1 & 4 & -4 & -4 & 0 \\ 2 & 9 & -9 & -9 & 0 \\ -3 & -7 & 7 & 7 & 0 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & -1 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] .$$
2. $$B$$ is not in reduced row echelon form because the pivots are not descending to the right.