1. Reduced Row Echelon Form
Three topics for the day:
Elementary Row Operations
Row Echelon Form
Reduced Row Echelon Form
…in Sec. 7.3b

2. When using Gaussian Elimination, we work primarily on the
coefficients of the variables. Knowing this, it is often easier to
perform these operations on the matrix form of a system of
equations…
Example – the system:
The augmented matrix:
The coefficient matrix:

3. Definition: Row Echelon Form (REF) of a Matrix
A matrix is in row echelon form if the following conditions are
satisfied:
1. Rows consisting entirely of 0’s (if there are any) occur at
the bottom of the matrix.
2. The first entry in any row with nonzero entries is 1.
3. The column subscript of the leading 1 entries increases as
the row subscript increases.
This should look
somewhat familiar…
Example:
Triangular Form?!?!

4. Elementary Row Operations on a Matrix
A combination of the following operations will transform a matrix
to row echelon form:
1. Interchange any two rows.
Notation: R indicates interchanging the i-th and j-th rows ij
2. Multiply all elements of a row by a nonzero real number.
Notation: kR indicates multiplying the i-th row by k i
3. Add a multiple of one row to any other row.
Notation: kR + R indicates adding k times the i-th row
to the j-th row i j

5. Example: Solve by Finding the REF
Augmented
Matrix:

6. Example: Solve by Finding the REF
z = 3
y – z = 4
 y = 7
x – y + 2z = –3
 x = –2
Solution: (x, y, z) = (–2, 7, 3)

7. 2R + R Guided Practice What elementary row operations applied to
will yield the given matrix?
2R + R 2 1

8. (1/4)R Guided Practice What elementary row operations applied to
will yield the given matrix?
(1/4)R 3

9. Guided Practice
Find a row echelon form for the given matrix.

10. Reduced Row Echelon Form

11. But first, a brief review of last few slides…
Augmented
Matrix:

12. Solution: (x, y, z) = (–1, 2, –3) z = –3 y – 2z = 8  y = 2
But first, a brief review of the last few slides…
z = –3
y – 2z = 8
 y = 2
x + 2y – 3z = 12
 x = –1
Solution: (x, y, z) = (–1, 2, –3)

13. Reduced Row Echelon Form
It is possible to continue applying elementary
row operations to a REF matrix until every
column that has a leading 1 has 0’s elsewhere.
This is the Reduced Row Echelon Form
(RREF) of the matrix.
Let’s see this with our previous example…

14. From this final RREF matrix, we can
read the solution of (–1, 2, –3) directly…
We can also use the
calculator to find
a RREF matrix…
 Try it with our initial augmented matrix:

15. More Good Examples Solution: (x, y, z, w) = (2, 7, –1, –3)
Solve the system using a RREF matrix
Augmented Matrix:
Solution:
(x, y, z, w) = (2, 7, –1, –3)
RREF:
(Calc)

16. More Good Examples Solution: Set of all ordered triples in the form
RREF Matrix:
Solve the system using a RREF matrix
For every value of z,
we can use these two
equations to find a
different x and y!!!
Solution: Set of all ordered triples in the form
(–3z + 5, 2z – 2, z), where z is any real number

17. More Good Examples Solution: Set of all ordered 4-tuples of the form
Solve the system using a RREF matrix
Augmented Matrix:
RREF Matrix:
Solution: Set of all ordered
4-tuples of the form
(–z – 2w + 1, 2z + w – 1, z, w), z and w any real #’s