4-2 Matrices and Systems of Linear Equations
Here is a system of 3 equations with 3 unknowns:
x - 2y + 3z
-x + 3y
2x - 5y + 5z
=9
= -4
= 17
(R1)
(R2)
(R3)
 we will not be writing such a system in the usual way
 there's an easier way to record and manipulate it . . .
 as a matrix (the augmented matrix of a system):
 1 2 3 9 


3 rows  1 3 0  4


 2  5 5 17 
4 columns
an example of a (3 x 4) matrix (3 rows by 4 columns).
A matrix is a rectangular array of numbers.
col
1
col
col
2
3

3
 5
1
row 1 

 =

1
3 
row 2   2
each element
has a name:
 a11

a
 21
a12
a 22
a13 

a 23 
row dimension: 2
column dimension: 3
dimensions of matrix: 2  3 ("two by three")
a square matrix has the same number of rows as columns
4-2
p. 1
Consider this system of equations (we will refer to the
equations as rows in accordance with the augmented matrix
representation of a system):
Row 1:
Row 2:
Row 3:
2x + 4y + 4z = 4
x + 3y + z = 4
-x + 3y + 2z = -1
1. If we interchange two rows, will that affect the
solution?
2. If we multiply a row by a constant, will that affect the
solution?
3. If we add one row to another, will that affect the
solution?
To answer questions 2 and 3, think about the strategies
used in solving by elimination in Section 4.1.
The answer to all the above questions is “No”. In fact, If
we combine 2 and 3 (add a multiple of one row to another),
that will also not alter the system’s solution.
These considerations lead to the following strategy:






start with an augmented matrix
perform a series of row operations on it
(they do not alter the solution of the system)
use a clever method, whose resulting augmented matrix
is one whose solution is easily obtained
thus yielding the solution to the original system
4-2
p. 2
Row operations
We will be doing elementary row operations on matrices
Remember: the system obtained by doing an elementary
row operation will have the same solution as the system
you started with.
Consider the following augmented matrix:
2 4 4 4
1 3 1 4


 1 3 2  1
Here are the row operations we will use:
1. Interchange (switch) two rows
Before the row operation:
2 4 4 4
1 3 1 4


 1 3 2  1
The row operation: R2  R3
After the row operation:
2 4 4 4
 1 3 2  1


 1 3 1 4 
This means:
(1) row 3 (R3) has replaced the old row 2 (R2)
(2) row 2 (R2) has replaced the old row 3 (R3)
4-2
p. 3
2. Multiply a row by a (non-zero) constant
Before the row operation:
2 4 4 4
1 3 1 4


 1 3 2  1
The row operation: 3R2  R2
2 4 4 4
 3 9 3 12 


 1 3 2  1
This means:
We have multiplied row 2 (R2) by 3 (thus replacing it).
This row operation always replaces a row by a (non-zero)
multiple of itself.
4-2
p. 4
3. Add a multiple of one row to another row
Before the row operation:
2 4 4 4
1 3 1 4


 1 3 2  1
The row operation: 2R1 + R3  R3
After the row operation:
 2 4 4 4
1 3 1 4 


3 11 ? ? 
 key row
 target row
This means: we have added 2 row 1 (R1) to row 3 (R3)
and replaced row 3 with that result.
We will refer to
 the first row mentioned (R1) as the key row
 the second row mentioned R3 as the target row
Row op 3 will always be of the form:
(constant)(key row) + (target row)  (target row)
Always: The second row mentioned will also be the row
that is pointed to (and that will be replaced).
4-2
p. 5