Gauss – Jordan Elimination
Lecture 6
Correction from last lecture (L5)
It was claimed that:
not every matrix can be changed
into reduced – row echelon form.
I have found no evidence to show that this is
true.
I am not sure if every matrix can or cannot be
changed into reduced row – echelon form.
My apologies =(
Gauss – Jordan elimination
In this lesson:
1. Changing a matrix to reduced row-echelon form.
2. Performing Gauss – Jordan elimination.
3. The difference between Gaussian elimination and
Gauss-Jordan elimination.
4. An augmented matrix with infinite solutions.
At the end of this lecture you should be able to:
1. Use elementary row operations to change a matrix into reduced rowechelon form.
2. Use elementary row operations to perform Gauss-Jordan elimination.
3. Explain the differences between Gaussian elimination (lecture 5) and Gauss
– Jordan elimination.
4. Explain how you can know if an augmented matrix will have infinite
solutions.
Changing a Matrix to Reduced Row – Echelon Form
Row-echelon form
Reduced row-echelon form
Reduced row-echelon form is unique. For every matrix
that can be reduced into row-echelon form, only one
reduction exists.
Changing a matrix to reduced row – echelon form:
3 3 3
−1 0 −4
2 4 −2
Changing a Matrix to Reduced Row – Echelon Form
Row-echelon form
Reduced row-echelon form
Reduced row-echelon form is unique. For every matrix
that can be reduced into row-echelon form, only one
reduction exists.
Changing a matrix to reduced row – echelon form:
3 3 3
−1 0 −4
2 4 −2
1
𝑅 →
3 1
𝟏 𝟏
−1 0
2 4
𝟏
−4
−2
Step 1:
Use row operations to
change the first row, first
column into a 1
Changing a Matrix to Reduced Row – Echelon Form
Row-echelon form
Reduced row-echelon form
Reduced row-echelon form is unique. For every matrix
that can be reduced into row-echelon form, only one
reduction exists.
Changing a matrix to reduced row – echelon form:
3 3 3
−1 0 −4
2 4 −2
1
𝑅 →
3 1
𝟏 𝟏
−1 0
2 4
𝟏
−4
−2
𝑅1 + 𝑅2 → 𝑅2
−2𝑅1 + 𝑅3 → 𝑅3
1 1
𝟎 𝟏
2 4
1
−𝟑
−2
1 1
0 1
𝟎 𝟐
1
−3
−𝟒
Step 1:
Use row operations to
change the first row, first
column into a 1
Step 2:
Use row multiplication
and addition to make all
the other numbers in that
column a 0
Changing a Matrix to Reduced Row – Echelon Form
Row-echelon form
Reduced row-echelon form
−𝑅2 + 𝑅1 → 𝑅1
Reduced row-echelon form is unique. For every matrix
that can be reduced into row-echelon form, only one
reduction exists.
Changing a matrix to reduced row – echelon form:
3 3 3
−1 0 −4
2 4 −2
1
𝑅 →
3 1
𝟏 𝟏
−1 0
2 4
𝟏
−4
−2
𝑅1 + 𝑅2 → 𝑅2
−2𝑅1 + 𝑅3 → 𝑅3
1 1
𝟎 𝟏
2 4
1
−𝟑
−2
1 1
0 1
𝟎 𝟐
1
−3
−𝟒
−2𝑅2 + 𝑅3 → 𝑅3
1
𝑅 →
2 3
3𝑅3 + 𝑅2 → 𝑅2
−4𝑅3 + 𝑅1 → 𝑅1
𝟏 𝟎
0 1
0 2
𝟒
−3
−4
1 0
0 1
𝟎 𝟎
4
−3
𝟐
1 0
0 1
𝟎 𝟎
4
−3
𝟏
1
𝟎
0
0 4
𝟏 𝟎
0 1
𝟏
0
0
𝟎 𝟎
1 0
0 1
Step 1:
Use row operations to
change the first row, first
column into a 1
Step 2:
Use row multiplication
and addition to make all
the other numbers in that
column a 0
Step 3:
Use row operations to
change the second row,
second column entry into
a 1.
Step 4:
Repeat step 2 then move
to column 3 and repeat
steps 1 and 2 until the
matrix is changed.
Start in the column on
the left, then move right.
As long as you work one
column at a time, you
will get it done.
Gauss – Jordan Elimination
Gauss – Jordan elimination uses reduced row – echelon form to solve a system of equations
Step 1:
Turn the system of
equations into an
augmented matrix
1
−1
2
−2
3
−5
3 ⋮
0 ⋮
5 ⋮
9
−4
17
Gauss – Jordan Elimination
Gauss – Jordan elimination uses reduced row – echelon form to solve a system of equations
1
−1
2
−2
3
−5
3 ⋮
0 ⋮
5 ⋮
9
−4
17
𝑅1 + 𝑅2 → 𝑅2
−2𝑅1 + 𝑅3 → 𝑅3
Try𝑅2changing
+ 𝑅3 → the
𝑅3
2𝑅
𝑅1
matrix
on
your
2+𝑅
1 →
1 check
own then
𝑅3 → 𝑅3
2
work
−3𝑅your
3 + 𝑅2 → 𝑅2
−9𝑅3 + 𝑅1 → 𝑅1
Step 1:
Turn the system of
equations into an
augmented matrix
Step 2:
Use elementary row
operations to change the
matrix into reduced row
– echelon form.
Gauss – Jordan Elimination
Gauss – Jordan elimination uses reduced row – echelon form to solve a system of equations
1
−1
2
−2
3
−5
3 ⋮
0 ⋮
5 ⋮
𝑅1 + 𝑅2 → 𝑅2
−2𝑅1 + 𝑅3 → 𝑅3
𝑅2 + 𝑅3 → 𝑅3
2𝑅2 + 𝑅1 → 𝑅1
1
𝑅 → 𝑅3
2 3
−3𝑅3 + 𝑅2 → 𝑅2
−9𝑅3 + 𝑅1 → 𝑅1
9
−4
17
Step 1:
Turn the system of
equations into an
augmented matrix
Step 2:
Use elementary row
operations to change the
matrix into reduced row
– echelon form.
Gauss – Jordan Elimination
Gauss – Jordan elimination uses reduced row – echelon form to solve a system of equations
1
−1
2
−2
3
−5
3 ⋮
0 ⋮
5 ⋮
𝑅1 + 𝑅2 → 𝑅2
−2𝑅1 + 𝑅3 → 𝑅3
𝑅2 + 𝑅3 → 𝑅3
2𝑅2 + 𝑅1 → 𝑅1
1
𝑅 → 𝑅3
2 3
−3𝑅3 + 𝑅2 → 𝑅2
−9𝑅3 + 𝑅1 → 𝑅1
9
−4
17
1 0
0 1
0 0
0 ⋮
0 ⋮
1 ⋮
1
−1
2
𝒙=𝟏
𝒚 = −𝟏
𝒛=𝟐
Step 1:
Turn the system of
equations into an
augmented matrix
Step 2:
Use elementary row
operations to change the
matrix into reduced row
– echelon form.
Step 3:
Change the reduced row
– echelon form matrix
back into a system of
equations. This will be
your answer.
Gauss – Jordan Elimination
Gauss – Jordan elimination uses reduced row – echelon form to solve a system of equations
1
−1
2
−2
3
−5
3 ⋮
0 ⋮
5 ⋮
𝑅1 + 𝑅2 → 𝑅2
−2𝑅1 + 𝑅3 → 𝑅3
𝑅2 + 𝑅3 → 𝑅3
2𝑅2 + 𝑅1 → 𝑅1
1
𝑅 → 𝑅3
2 3
−3𝑅3 + 𝑅2 → 𝑅2
−9𝑅3 + 𝑅1 → 𝑅1
9
−4
17
1 0
0 1
0 0
x y
0
0
1
z
⋮ 1
⋮ −1
⋮ 2
= ans
𝒙=𝟏
𝒚 = −𝟏
𝒛=𝟐
Step 1:
Turn the system of
equations into an
augmented matrix
Step 2:
Use elementary row
operations to change the
matrix into reduced row
– echelon form.
Step 3:
Change the reduced row
– echelon form matrix
back into a system of
equations. This will be
your answer.
Gaussian Elimination Vs. Gauss – Jordan Elimination
Both Gaussian Elimination and Gauss – Jordan Elimination use matrices to solve systems of
equations. They do, however, have slight differences.
Gaussian Elimination
• Uses row – echelon form.
• Has less work to reduce the matrix,
but more work to back substitute
once the reduction is finished.
• This is more useful when
programming computers
Gauss – Jordan Elimination
• Uses reduced row – echelon form.
• Has more work to reduce the
matrix, but less work to back
substitute once the reduction is
finished.
• This is more useful when finding
matrix inverses.
A System with Infinitely Many Solutions
When a system of equation has fewer equations than variables, it either has no solutions
or infinitely many solutions.
Since we cannot solve for z, what x and y equals
depending on what value we choose for z.
Since we can choose an infinite number of values for z,
we have an infinite number of solutions to this system
of equations.
After changing the matrix to row – echelon form:
This creates the system of equations:
The problem with this is that we can never solve for z,
we don’t have a way of eliminating 2 out of the 3
variables in this system.
When we look at coefficient matrix:
𝟏 𝟎 𝟓
𝟎 𝟏 −𝟑
we can already see that this equation will have an
infinite number of solutions because the matrix has 3
variable columns and only 2 equation rows.
If a system of equation’s coefficient
matrix has more columns than rows, then
the matrix may have infinitely many
solutions.