4. Matrices and Systems Of Linear Equations
A matrix is simply a rectangular array of numbers. Matrices
are used to organize information into categories that
correspond to the rows and columns of the matrix.
For example, a scientist might organize information on a
population of endangered whales as follows:
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5. Matrices and Systems Of Linear Equations
In this section we represent a linear system by a matrix,
called the augmented matrix of the system:
The augmented matrix contains the same information as
the system, but in a simpler form.
The operations we learned for solving systems of equations
can now be performed on the augmented matrix.
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10. The Augmented Matrix of a Linear System
We can write a system of linear equations as a matrix,
called the augmented matrix of the system, by writing only
the coefficients and constants that appear in the equations.
Here is an example.
Linear system Augmented matrix
Notice that a missing variable in an equation corresponds
to a 0 entry in the augmented matrix.
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11. Write the augmented matrix of the system of equations.
Solution:
First we write the linear system with the variables lined up
in columns.
Example 1 – Finding the Augmented Matrix of a Linear System
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12. Example 1 – Solution
The augmented matrix is the matrix whose entries are the
coefficients and the constants in this system.
cont’d
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14. Elementary Row Operations
Note that performing any of these operations on the
augmented matrix of a system does not change its solution. 14
15. Elementary Row Operations
We use the following notation to describe the elementary
row operations:
Symbol
Ri + kRj Ri
kRi
Ri Rj
Description
Change the ith row by adding k
times row j to it, and then put the
result back in row i.
Multiply the ith row by k.
Interchange the ith and jth rows.
In the next example we compare the two ways of writing
systems of linear equations.
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16. Example 2 – Using Elementary Row Operations to Solve a Linear System
Solve the system of linear equations.
Solution:
Our goal is to eliminate the x-term from the second
equation and the x- and y-terms from the third equation.
For comparison, we write both the system of equations and
its augmented matrix.
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17. Example 2 – Solution
System Augmented matrix
cont’d
Add (–1)
Equation 1 to
Equation 2.
Add (–3)
Equation 1
to Equation 3.
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18. Example 2 – Solution
System Augmented matrix
Multiply
Equation 3
by .
Add (–3)
Equation 3
to Equation 2
(to eliminate y
from Equation 2).
R2 – 3R3 R2
cont’d
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0 3 -5 6
19. Example 2 – Solution
System Augmented matrix
Now we use back-substitution to find that x =2, y =7, and
z =3. The solution is (2, 7, 3).
cont’d
Interchange
Equations
2 and 3.
R2 R3
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