This document discusses matrices and systems of linear equations. It begins by defining matrices and their dimensions. It then explains how a system of linear equations can be represented by an augmented matrix containing the coefficients and constants. The document covers elementary row operations that can be performed on augmented matrices to solve systems of linear equations, such as adding multiples of rows and multiplying rows by constants. It provides examples of writing augmented matrices from word problems and using row operations to solve systems. The objectives are to understand matrices, augmented matrices, elementary row operations, and how to use them to solve systems of linear equations.

1. Matrices and Systems Of
Linear Equations
Prepared by:
WANNY B. JARON

2. Objectives
2
► Matrices
► The Augmented Matrix of a Linear System
► Elementary Row Operations
► Gaussian Elimination

3. Objectives
3
► Gauss-Jordan Elimination
► Inconsistent and Dependent Systems
► Modeling with Linear Systems

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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6. Matrices
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7. Matrices
We begin by defining the various elements that make up a
matrix.
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8. Matrices
Here are some examples of matrices.
Matrix Dimension
2  3
[6 –5 0 1] 1  4
2 rows by 3 columns
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1 row by 4 columns

9. The Augmented Matrix
of a Linear System
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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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13. Elementary Row Operations
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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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20. Practice Exercises:
Find the solution set of the ff system of equation using
elementary row operations.