Drexel University  ENGR 231 Linear Engineering Systems
Copyright 2011 Drexel University
Page 1
Lab 2: Matrix Manipulation and Elementary Row Reduction
Oleh Tretiak and Donald Bucci
Goals for Lab 2:
1.
Basic Matrix Manipulation Techniques
2.
Manual Row Reduction and the RREF Function
Prelab Exercise – Manual and Automated Row Reduction
Part 1 – Introduction and Motivation
One of the key functionalities of MATLAB is its ability to efficiently handle vector and
matrix operations. In this lab, we introduce basic matrix manipulation techniques through the
conversion of a system into
reduced row echelon form
(RREF)
.
Let’s consider row reduction of the following system of linear equations:
x
1
+
2
x
2
+
3
x
3
=
4
4
x
1
+
5
x
2
+
6
x
3
=
7
6
x
1
+
7
x
2
+
8
x
3
=
9
We can rewrite this as an
augmented matrix
, resulting in:
1
2
3
4
4
5
6
7
6
7
8
9
!
"
#
#
#
#
$
%
&
&
&
&
Our goal is to row reduce the above system in an attempt to find its solution set. Recall that
this can be obtained by performing elementary row operations until the system is first in
echelon
form
or
triangular form
(i.e. all entries in a column below leading entries are zero). This may, in
some cases, look like the following:
#
#
#
#
0
#
#
#
0
0
#
#
!
"
#
#
#
#
$
%
&
&
&
&
Once arriving at echelon form, we can continue to the RREF (i.e. the leading entry in each row is
equal to one and is the only nonzero entry in each column). This may also, in some cases, look
like the following:
1
0
0
#
0
1
0
#
0
0
1
#
!
"
#
#
#
#
$
%
&
&
&
&
We now show how to perform these operations in MATLAB.
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Drexel University  ENGR 231 Linear Engineering Systems
Copyright 2011 Drexel University
Page 2
Part 2 – Performing the Operations in MATLAB
Firstly, we need to define
a MATLAB varia
ble
for the augmented matrix. This is done with
the following statement:
myMatrix = [1 2 3 4; 4 5 6 7; 6 7 8 9]
This is another version of the bracket statement that we used in the first lab. Notice that matrix
generation within MATLAB is handled one row at a time. The first row is made by using spaces
(or commas) to delineate sequential terms. Once a row is completed, a
semicolon
signifies the
beginning of another row, and so on.
For ease of explanation, let’s define any position within the matrix as (
r,c
), where
r
and
c
represents the row and column indices of the specified matrix element. Our next step is to make
locations (2,1) and (3,1) each equal to zero. This may be accomplished by performing the
following elementary row operations:
STEP 1.
R
2
←
4R
1
+ R
2
STEP 2.
R
3
←
6R
1
+ R
3
We thus need to know how to
select an entire row out of a variable
in MATLAB. To do this, we
use the following syntax:
variableName(row#,:)
where
row
#
delineates the selected row within the matrix defined in the variable whose name is
variableName
. Thus, the commands that perform the above two steps are as follows:
myMatrix(2,:) =  4*myMatix(1,:) + myMatix(2,:)
myMatrix(3,:) =  6*myMatix(1,:) + myMatix(3,:)
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 Fall '10
 OlehTretiak
 Row, Linear Engineering Systems

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