Overview
Matrix Solutions
to Linear Systems
Section 8.1
Three-variable systems
• A system of linear equations in three
variables is in the form
• When solving systems of linear equations
in two variables, we utilized the following
techniques:
1.Substitution
2.Elimination
3.Graphing
• In this section we will develop techniques
that can be used for larger systems.
Solving a System of 3 Equations
with 3 Variables
• Graphically, you are attempting to find
where 3 planes intersect.
• If you find 3 numeric values for (x,y,z), this
indicates the 3 planes intersect at that
point.
• The solution to a three-variable system is
an ordered triple (x,y,z).
Solving a three-variable system
• We will utilize matrix techniques.
• One method involves using row
operations, and is done by hand.
• The other method involves using your
graphing calculator.
Matrices
• Matrices are a valuable tool when used to solve systems
of linear equations.
• A matrix is a rectangular array of numbers.
• The rows of a matrix are horizontal.
• The columns of a matrix are vertical.
• The matrix shown has 2 rows and 3 columns
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Systems of Equations as Matrices
• We can put linear equations into augmented matrices
and perform row operations corresponding to the
equation manipulations and combinations we have done
in the past.
The augmented matrix
• If
is a system of linear
equations in three variables, then
is the augmented matrix for the system.
Example 1
Example 2
Row Operations
Row Operations
• When you have a matrix, there are
operations you can perform on the rows in
that matrix:
1.You can swap rows:
2.You can multiply a row by a constant (and
put the result in the place of the original):
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Row Operations
Row Operations
3.You can add two rows together and put
the result in the place of one of the rows you
added (the other row is unchanged):
The Goal Of Row Operations
• Use row operations to convert your
augmented matrix to row echelon form:
4. You can do combinations:
Example 3
Which of the following matrices are in row-echelon form?
a)
b)
c)
d)
• On the next slide, you’ll identify this form
and see why row echelon form is useful.
Getting There Is Half The Fun
• How to convert an Augmented Matrix into
row echelon form:
1.Start working with the first column. If there
is a “1” at the top of that column, go to
Step 2. If there is not a “1” at the top of
that column, swap that row with a row that
has a “1” in the first position.
2. The next objective is get a “0” in the
remaining positions of the first column.
Multiply Row 1 by the opposite of the
coefficient of the first entry in the second
row, then add to Row 2 and put the result
in Row 2.
3. Repeat the process with Row 3, again
using Row 1.
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4. Now move on to the second column. You
want to get a zero in the last position. If
there is a “1” in the second position of the
second column, multiply Row 2 by the
opposite of the entry in the second
position of Row 3, then add to Row 3 and
put the result in Row 3.
5. If there is not a “1” in the second position
of the second column, you will have to be
a bit more “creative”.
Example 4
Perform the matrix row operation and write the new matrix.
Example 5
Gaussian Elimination with Matrices
• Row-Equivalent Operations
1. Interchange any two rows.
2. Multiply each entry in a row by the same
nonzero constant.
3. Add a nonzero multiple of one row to
another row.
We can use matrices, applying Gaussian
Elimination, to solve linear systems.
Gauss-Jordan Elimination
Solve the following system using Gaussian elimination:
• Continues the process until there are 1s on the main
diagonal and 0s below and above the main diagonal.
• Such a matrix is said to be in reduced row-echelon
form.
• Either method can be applied to all of our problems.
• We will just focus on Gaussian Elimination in this
course when work is to be done by hand.
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Gauss-Jordan Elimination Example
Gauss-Jordan Elimination continued
We continue to perform
row-equivalent operations
until we have a matrix in
reduced row-echelon form.
• Example: Use
Gauss-Jordan
elimination to solve
the system of
equations from the
previous example.
Gauss-Jordan Elimination continued
Next, we multiply the
second row by 3 and
add it to the first row.
Gauss-Jordan Elimination continued
• Writing the system of
equations that
corresponds to this
matrix, we have
We can actually read the solution, (2, −1, 3), directly
from the last column of the reduced row-echelon matrix.
If this method (Gauss-Jordan) is quicker,
why would we use Gaussian elimination?
• Using a matrix to solve a system by
Gaussian elimination provides a standard,
programmable approach.
• When computer programs (which may be
contained in calculators) solve systems,
this is the method utilized!
Example 6
Solve the system of equations using Gaussian elimination.
3y – z = - 1
x + 5y - z = - 4
-3x + 6y + 2z = 11
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