8.5. Row Operations and Row Echelon Forms http://www.ck12.org
8.5 Row Operations and Row Echelon Forms
Here you will manipulate matrices using row operations into row echelon form and reduced row echelon form.
Applying row operations to reduce a matrix is a procedural skill that takes lots of writing, rewriting and careful
arithmetic. The payoff for being able to transform a matrix into a simplified form will become clear later. For now,
what does the simplified form mean for a matrix?
Watch This
MEDIA
Click image to the left for use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/61454http://www.youtube.com/watch?v=LsnOlNjWWug James Sousa: Introduction to Augmented Matrices
Guidance
There are only three operations that are permitted to act on matrices. They are the exact same operations that are
permitted when solving a system of equations.
- Add a multiple of one row to another row.
- Scale a row by multiplying through by a non-zero constant.
- Swap two rows.
Using these three operations, your job is to simplify matrices intorow echelon form. Row echelon form must meet
three requirements.
- The leading coefficient of each row must be a one.
- All entries in a column below a leading one must be zero.
- All rows that just contain zeros are at the bottom of the matrix.
Here are some examples of matrices in row echelon form:
[1 14
0 1]
,
[1 2 3
0 1 4
]
,
1 2 3 5 6
0 0 1 4 7
0 0 0 1 − 2
0 0 0 0 0
Reduced row echelon formalso has one extra stipulation compared with row echelon form.
- Every leading coefficient of 1 must be the only non-zero element in that column.
Here are some examples of matrices in reduced row echelon form: