The Gaussian Elimination, is a method of putting a matrix in row echelon form (REF), using elementary row operations.
REF is when a matrix qualifies for the following two characteristics:
- Each nonzero row has a leading coefficient (the first nonzero entry) that is to the right of the leading coefficient of the row above it
- There can’t be any nonzero entries below the leading coefficient in the leading coefficient’s column
- All zero rows are at the bottom
Note: A matrix in REF is not unique, so you may have a slightly different solution.
Here are examples of REF:
Here are examples of matrices that aren’t in REF:
- Switch any two rows
- Multiply each entry in a row by a non-zero constant
- Replace a row by the sum/difference of the row itself and another row, where it’s entries are multiplied by a non-zero constant
Here is a guideline on how to put a matrix in REF:
- Move all zero rows to the bottom
- Begin at the first row
- If the first entry is a zero, switch the row with a row below it that has non-zero entry in the first column
- If there are nonzero entries below the leading coefficient of the first row in the same column, cancel the entries by subtracting multiples of the the first row to the other rows (this will result in a zero entry)
- Repeat steps 2 - 4 and replace “first” with “second”, then “third”, and so on until you can’t do anything more
This topic can be hard to understand at first, so let’s see some examples to better understand.
First example (click here):
Next example (click here):
Putting matrices in REF can be tricky at first, but once you’ve done a handful of practice problems, it will come to you much easier. Check out Symbolab’s Row Echelon Calculator to help you better understand this topic.
Until next time,
Leah