Row Echelon Form
Row Echelon - How it Works - Video
What is Row Echelon Form?
What is Row Echelon Form?
Row echelon form has the following 3 rules. The first rule is the first non-zero element is each row, called the leading entry is 1. (Some textbooks say different. Ask your teacher.) The second rule is that each leading entry is in a column to the right of the leading entry in the previous row. The the third rule is that rows with all zero elements, if any, are below rows having a non-zero element.
Row Echelon Form - Rule 1
Row Echelon Form - Rule 1
In the image above, the principal diagonal is full of 1s and 0s.
Row Echelon Form - Rule 2
Row Echelon Form - Rule 2
In the image above, the numbers below the principal diagonal are 0s.
Row Echelon Form - Rule 3
Row Echelon Form - Rule 3
R4 has only 0s so it is the last row.
Reduced Row Echelon Form
Reduced Row Echelon Form:
Reduced row echelon form is the same as row echelon form, but with one extra facet. And that is that the numbers above the principal diagonal are always 0.
We have created the identity matrix. We can use Gauss-Jordan Elimination to find an inverse an invertible matrix.
Row Operations
Guidelines For Finding the Echelon Form
1. Locate the first column with the number 1 and transform it by any of the 3 row operations most likely operation 1 or 2.
2. Apply operation 3 to get 0s under the number and the remaining rows.
3. Disregard the first row. Locate the next column that contains nonzero elements and transform it by applying any of the 3 row operations most likely 1 or 2 to get 1 in the next column and in the second row.
4. Apply operation 3 to get 0s under the number and the remaining rows.
5. Disregard the first row. Locate the next column that contains nonzero elements and transform it by applying any of the 3 row operations most likely 1 or 2 to get 1 in the next column and in the second row.
6. Continue the process until either row echelon from or reduced row echelon is formed.
Sometimes the numbers are not friendly (fractions will be involved here) and when that happens. I like to do is to leave the first nonzero element in each row as is. For instance, if it is 4, then don't multiply by 1/4 until the last. This way the fractions don't appear until the last step. The numbers might be a little bigger, but there won't be any fractions.
Live Worksheet
Here is the link if you prefer.