03 Row Echelon Form
A matrixis in Row Echelon Form if:
- The leading entry (first non-zero number from left to right) in any row is to the right of the leading entries above it.
- Every entry in a column below a leading entry is zero.
- If there is a zero row, it is below all other rows.
Notice the “staircase” like form of the matrix.
2, 2, and 4 are considered leading entries.
Reduced Row Echelon Form (rref)
Matrices are in Reduced Row Echelon Form (RREF) if they are in Row Echelon Form, AND the following:
- Every leading Entry is 1
- Every entry above and below any leading entries is 0
$$ \left[\begin{array}{ccc|c} \fbox{1} & 0 & 0 & 3 \\ 0 & \fbox{1} & 0 & -4 \\ 0 & 0 & \fbox{1} & 5 \end{array}\right] \text{ is in rref.} $$Every matrix has a one unique rref form.
They can look weird too:
$$ \left[\begin{array}{cccc|c} \fbox{1} & -2 & 0 & 3 & 4 \\ 0 & 0 & \fbox{1} & 2 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array}\right] \text{ is also in rref.} $$The leading entries are boxed above. They are also called pivots.
Reduced Row Echelon Form makes it extremely easy to see the solutions to a system, keeping in mind that
All we have to do is go column by column and isolate the variables that are in pivot columns, and leave all other variables as
Tip: If two augmented matrices have the same coefficient matrices, you can use the same operations to row reduce them.