Upload
phungngoc
View
234
Download
0
Embed Size (px)
Citation preview
Gauss – Jordan EliminationLecture 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 row-
echelon 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.
3 3 3−1 0 −42 4 −2
Changing a matrix to reduced row – echelon form:
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.
3 3 3−1 0 −42 4 −2
𝟏 𝟏 𝟏−1 0 −42 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:
1
3𝑅1 →
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.
3 3 3−1 0 −42 4 −2
𝟏 𝟏 𝟏−1 0 −42 4 −2
1 1 1𝟎 𝟏 −𝟑2 4 −2
1 1 10 1 −3𝟎 𝟐 −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:
1
3𝑅1 →
𝑅1 + 𝑅2 → 𝑅2
−2𝑅1 + 𝑅3 → 𝑅3
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.
3 3 3−1 0 −42 4 −2
𝟏 𝟏 𝟏−1 0 −42 4 −2
1 1 1𝟎 𝟏 −𝟑2 4 −2
1 1 10 1 −3𝟎 𝟐 −𝟒
𝟏 𝟎 𝟒0 1 −30 2 −4
1 0 40 1 −3𝟎 𝟎 𝟐
1 0 40 1 −3𝟎 𝟎 𝟏
1 0 4𝟎 𝟏 𝟎0 0 1
𝟏 𝟎 𝟎0 1 00 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
Changing a matrix to reduced row – echelon form:
1
3𝑅1 →
𝑅1 + 𝑅2 → 𝑅2
−2𝑅1 + 𝑅3 → 𝑅3
−𝑅2 + 𝑅1 → 𝑅1
−2𝑅2 + 𝑅3 → 𝑅3
1
2𝑅3 →
3𝑅3 + 𝑅2 → 𝑅2
−4𝑅3 + 𝑅1 → 𝑅1
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 EliminationGauss – Jordan elimination uses reduced row – echelon form to solve a system of equations
1 −2 3 ⋮ 9−1 3 0 ⋮ −42 −5 5 ⋮ 17
Step 1:Turn the system of equations into an augmented matrix
Gauss – Jordan Elimination
Step 1:Turn the system of equations into an augmented matrix
Gauss – Jordan elimination uses reduced row – echelon form to solve a system of equations
1 −2 3 ⋮ 9−1 3 0 ⋮ −42 −5 5 ⋮ 17
𝑅1 + 𝑅2 → 𝑅2−2𝑅1 + 𝑅3 → 𝑅3
𝑅2 + 𝑅3 → 𝑅32𝑅2 + 𝑅1 → 𝑅1
1
2𝑅3 → 𝑅3
−3𝑅3 + 𝑅2 → 𝑅2−9𝑅3 + 𝑅1 → 𝑅1
Try changing the matrix on your own then check
your work
Step 2:Use elementary row operations to change the matrix into reduced row – echelon form.
Gauss – Jordan Elimination
Step 1:Turn the system of equations into an augmented matrix
Gauss – Jordan elimination uses reduced row – echelon form to solve a system of equations
1 −2 3 ⋮ 9−1 3 0 ⋮ −42 −5 5 ⋮ 17
𝑅1 + 𝑅2 → 𝑅2−2𝑅1 + 𝑅3 → 𝑅3
𝑅2 + 𝑅3 → 𝑅32𝑅2 + 𝑅1 → 𝑅1
1
2𝑅3 → 𝑅3
−3𝑅3 + 𝑅2 → 𝑅2−9𝑅3 + 𝑅1 → 𝑅1
Step 2:Use elementary row operations to change the matrix into reduced row – echelon form.
Gauss – Jordan Elimination
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 uses reduced row – echelon form to solve a system of equations
1 −2 3 ⋮ 9−1 3 0 ⋮ −42 −5 5 ⋮ 17
𝑅1 + 𝑅2 → 𝑅2−2𝑅1 + 𝑅3 → 𝑅3
𝑅2 + 𝑅3 → 𝑅32𝑅2 + 𝑅1 → 𝑅1
1
2𝑅3 → 𝑅3
−3𝑅3 + 𝑅2 → 𝑅2−9𝑅3 + 𝑅1 → 𝑅1
1 0 0 ⋮ 10 1 0 ⋮ −10 0 1 ⋮ 2
𝒙 = 𝟏𝒚 = −𝟏𝒛 = 𝟐
Gauss – Jordan Elimination
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 uses reduced row – echelon form to solve a system of equations
1 −2 3 ⋮ 9−1 3 0 ⋮ −42 −5 5 ⋮ 17
𝑅1 + 𝑅2 → 𝑅2−2𝑅1 + 𝑅3 → 𝑅3
𝑅2 + 𝑅3 → 𝑅32𝑅2 + 𝑅1 → 𝑅1
1
2𝑅3 → 𝑅3
−3𝑅3 + 𝑅2 → 𝑅2−9𝑅3 + 𝑅1 → 𝑅1
1 0 0 ⋮ 10 1 0 ⋮ −10 0 1 ⋮ 2x y z = ans
𝒙 = 𝟏𝒚 = −𝟏𝒛 = 𝟐
Gaussian Elimination Vs. Gauss – Jordan EliminationBoth Gaussian Elimination and Gauss – Jordan Elimination use matrices to solve systems of equations. They do, however, have slight differences.
Gaussian Elimination Gauss – Jordan 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
• 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 SolutionsWhen a system of equation has fewer equations than variables, it either has no solutionsor infinitely many solutions.
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.
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.
When we look at coefficient matrix:𝟏 𝟎 𝟓𝟎 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.