# 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 * free*, meaning they can be anything. It might be helpful to add that variables that are not free are called

basic variables.

If two augmented matrices have the same coefficient matrices, you can use the same operations to row reduce them.Tip: