Reduction to reduced row echelon form
Here’s a bit of info and examples on Reduced Row Echelon Form (RREF).
The definition
We say that a matrix is in reduced row echelon form if the following properties are satisfied.
- If the entries in a row are all zero, then the same is true of any row below it.
- If we move across a row from left to right, the first nonzero entry we encounter is 1. We call this entry the leading entry in the row.
- The leading entry in any row is to the right of the leading entries in all the rows above it.
- A leading entry is the only nonzero entry in its column.
Some RREF examples??
Which of these look like they are in reduced row echelon form?
\[ B_1=\begin{bmatrix} 1&0&2&0&0\\ 0&1&-3&0&0\\ 0&0&0&1&5\\ 0&0&0&0&0 \end{bmatrix} \]
\[ B_2=\begin{bmatrix} 1&0&2&0&0\\ 0&0&0&0&0\\ 0&1&-3&0&0\\ 0&0&0&1&5 \end{bmatrix} \]
\[ B_3=\begin{bmatrix} 1&2&0&0&1\\ 0&1&0&0&-3\\ 0&0&0&1&4\\ 0&0&0&0&0 \end{bmatrix} \]
Performing row reduction
Here area few examples illustrating row reduction to reduced row echelon form.
The typical situation
\[ \begin{aligned} \left[ \begin{array}{rrr|r} 1&0&1&2\\ 0&0&1&3\\ 1&1&0&-1 \end{array} \right] &\xrightarrow{\;R_3\leftarrow R_3-R_1\;} \left[ \begin{array}{rrr|r} 1&0&1&2\\ 0&0&1&3\\ 0&1&-1&-3 \end{array} \right] \\[1.0em] &\xrightarrow{\;R_2\leftrightarrow R_3\;} \left[ \begin{array}{rrr|r} 1&0&1&2\\ 0&1&-1&-3\\ 0&0&1&3 \end{array} \right] \\[1.0em] &\xrightarrow{\;R_2\leftarrow R_2+R_3\;} \left[ \begin{array}{rrr|r} 1&0&1&2\\ 0&1&0&0\\ 0&0&1&3 \end{array} \right] \\[1.0em] &\xrightarrow{\;R_1\leftarrow R_1-R_3\;} \left[ \begin{array}{rrr|r} 1&0&0&-1\\ 0&1&0&0\\ 0&0&1&3 \end{array} \right] \end{aligned} \]
Infinitely many solutions
\[ \begin{aligned} \left[ \begin{array}{rrr|r} 1&2&3&1\\ 4&5&6&1\\ 7&8&9&1 \end{array} \right] &\xrightarrow{\;R_2\leftarrow R_2-4R_1,\; R_3\leftarrow R_3-7R_1\;} \left[ \begin{array}{rrr|r} 1&2&3&1\\ 0&-3&-6&-3\\ 0&-6&-12&-6 \end{array} \right] \\[1.2em] &\xrightarrow{\;R_2\leftarrow -\tfrac{1}{3}R_2\;} \left[ \begin{array}{rrr|r} 1&2&3&1\\ 0&1&2&1\\ 0&-6&-12&-6 \end{array} \right] \\[1.2em] &\xrightarrow{\;R_3\leftarrow R_3+6R_2\;} \left[ \begin{array}{rrr|r} 1&2&3&1\\ 0&1&2&1\\ 0&0&0&0 \end{array} \right] \\[1.2em] &\xrightarrow{\;R_1\leftarrow R_1-2R_2\;} \left[ \begin{array}{rrr|r} 1&0&-1&-1\\ 0&1&2&1\\ 0&0&0&0 \end{array} \right] \end{aligned} \]
An inconsistent system
\[ \begin{aligned} \left[ \begin{array}{rrr|r} 1&4&5&7\\ 2&8&10&14\\ 3&12&15&22 \end{array} \right] &\xrightarrow{\;R_2\leftarrow R_2-2R_1,\; R_3\leftarrow R_3-3R_1\;} \left[ \begin{array}{rrr|r} 1&4&5&7\\ 0&0&0&0\\ 0&0&0&1 \end{array} \right] \\[1.0em] &\xrightarrow{\;R_2\leftrightarrow R_3\;} \left[ \begin{array}{rrr|r} 1&4&5&7\\ 0&0&0&1\\ 0&0&0&0 \end{array} \right] \\[1.0em] &\xrightarrow{\;R_1\leftarrow R_1-7R_2\;} \left[ \begin{array}{rrr|r} 1&4&5&0\\ 0&0&0&1\\ 0&0&0&0 \end{array} \right] \end{aligned} \]
A randomly chosen system
\[ \begin{aligned} \left[ \begin{array}{rrr|r} 8 & 12 & 10 & 7\\ 12 & 9 & 1 & 10\\ 5 & 9 & 1 & 2 \end{array} \right] &\xrightarrow{\;R_1 \leftrightarrow R_2\;} \left[ \begin{array}{rrr|r} 12 & 9 & 1 & 10\\ 8 & 12 & 10 & 7\\ 5 & 9 & 1 & 2 \end{array} \right] \\[1.2em] &\xrightarrow{\;R_2\leftarrow R_2-\tfrac{2}{3}R_1,\; R_3\leftarrow R_3-\tfrac{5}{12}R_1\;} \left[ \begin{array}{rrr|r} 12 & 9 & 1 & 10\\ 0 & 6 & \tfrac{28}{3} & \tfrac{1}{3}\\ 0 & \tfrac{21}{4} & \tfrac{7}{12} & -\tfrac{13}{6} \end{array} \right] \\[1.2em] &\xrightarrow{\;R_2\leftarrow \tfrac{1}{6}R_2\;} \left[ \begin{array}{rrr|r} 12 & 9 & 1 & 10\\ 0 & 1 & \tfrac{14}{9} & \tfrac{1}{18}\\ 0 & \tfrac{21}{4} & \tfrac{7}{12} & -\tfrac{13}{6} \end{array} \right] \\[1.2em] &\xrightarrow{\;R_1\leftarrow R_1-9R_2,\; R_3\leftarrow R_3-\tfrac{21}{4}R_2\;} \left[ \begin{array}{rrr|r} 12 & 0 & -13 & \tfrac{19}{2}\\ 0 & 1 & \tfrac{14}{9} & \tfrac{1}{18}\\ 0 & 0 & -\tfrac{91}{12} & -\tfrac{59}{24} \end{array} \right] \\[1.2em] &\xrightarrow{\;R_3\leftarrow -\tfrac{12}{91}R_3\;} \left[ \begin{array}{rrr|r} 12 & 0 & -13 & \tfrac{19}{2}\\ 0 & 1 & \tfrac{14}{9} & \tfrac{1}{18}\\ 0 & 0 & 1 & \tfrac{59}{182} \end{array} \right] \\[1.2em] &\xrightarrow{\;R_2\leftarrow R_2-\tfrac{14}{9}R_3,\; R_1\leftarrow R_1+13R_3\;} \left[ \begin{array}{rrr|r} 12 & 0 & 0 & \tfrac{96}{7}\\ 0 & 1 & 0 & -\tfrac{35}{78}\\ 0 & 0 & 1 & \tfrac{59}{182} \end{array} \right] \\[1.2em] &\xrightarrow{\;R_1\leftarrow \tfrac{1}{12}R_1\;} \left[ \begin{array}{rrr|r} 1 & 0 & 0 & \tfrac{8}{7}\\ 0 & 1 & 0 & -\tfrac{35}{78}\\ 0 & 0 & 1 & \tfrac{59}{182} \end{array} \right] \end{aligned} \]