Matrices and MATLAB
Dr Viktor Fedun
Automatic Control and Systems
Engineering, C09
Based on lectures by Dr Anthony Rossiter
Examples of a matrix
Examples of a matrix
Examples of a matrix
A matrix can be thought of simply as a table
of numbers with a given number of rows
and columns.
Example A has 3 rows and 3 columns,
example B has 4 rows and 4 columns.
 2

A 5

  3
0
6
1
3

9 ;

4 
2

4

B 
7

 54
3
4
5
2
8
6
23
1
6 

5

8 

2 
Examples of
square
matrices
Examples of a matrix
A matrix can be thought of simply as a table
of numbers with a given number of rows
and columns.
Example A has 2 rows and 3 columns,
example B has 3 rows and 1 column.
2
A 
 1
3
0
4
;
5
5 
 
B  6
 
 7 
Examples of
non-square
matrices
What use are matrices?
• Compact form for storing data (equivalent to
a table but more flexible).
• Each column(or row) can relate to a different
‘state’: e.g. Time, displacement, velocity,
temperature, etc.
• Convenient for computer code (and algebra)
as can store and share large quantities of
related data with a single variable.
• Easily extend to dynamic relations/modelling.
Indexing of matrices
Always give as {row index,column index}
Start counting from top left, so the top left
component is index {1,1}, e.g. indices are
given as:
Column
Row index
increasing
 {1,1}

{ 2 ,1}

 {3 ,1}

{ 4 ,1}
{1, 2}
{1, 3}
{ 2 , 2}
{ 2 , 3}
{3 , 2}
{3 , 3}
{ 4 , 2}
{ 4 , 3}
{1, 4} 

{ 2 , 4}

{3 , 4} 

{ 4 , 4} 
index
increasing
What is the index of the
component -4
1.
2.
3.
4.
5.
{1,3}
{1,2}
{2,1}
{3,1}
{4,3}
 2

4

B 
 7

 54
3
4
5
2
8
6
23
1
6 

5

8 

2 
What is the index of the
component -4
1.
2.
3.
4.
5.
{1,3}
{1,2}
{2,1}
{3,1}
{4,3}
 2

4

B 
 7

 54
 {1,1}

{ 2 ,1}

 {3 ,1}

{ 4 ,1}
3
4
5
2
8
6
23
1
{1, 2}
{1, 3}
{ 2 , 2}
{ 2 , 3}
{3 , 2}
{3 , 3}
{ 4 , 2}
{ 4 , 3}
6 

5

8 

2 
{1, 4} 

{ 2 , 4}

{3 , 4} 

{ 4 , 4} 
Create a matrix with following
information
2nd row and 3rd column is 5
4th row and 2nd column is 2
1st row and 5th column is 6
Remainder are 0.
TRY this on MATLAB
A(2,3)=5; A(4,2)=2; A(1,5)=6;
disp(A)
Note
MATLAB
syntax
matches that
used in
mathematics,
i.e. row index
and then
column
index.
Extracting
parts of
matrices
Example of
selecting given
rows and
columns.
Can rearrange the
order if desired.
Matrices
The main thing to note is that the default
variable in MATLAB is a matrix (or vector if
the row or column dimension is one).
Any BODMAS type operation that is valid
with matrices can be carried out with the
same syntax.
MATLAB also includes a large number of
matrix analysis tools that you will find
useful in the 2nd year.
Matrices and MATLAB for plots
A student collects data from an experiment
and stores the answer in a matrix data_exp1.
• Column 1 corresponds to time.
• Column 2 to temperature
• Column 3 is voltage supplied.
It is now simple to code and plot this data – the
corresponding code is shown next.
MATLAB
example
30
25
20
Temperature
Voltage
15
10
5
0
0
20
40
60
Time (sec)
80
100
120
Correcting data
Student is told that the temperature readings
were all out by 2 degrees due to a
calibration error.
Once the data is in a matrix, update is very
efficient.
Use line: data_exp1(:,2)=data_exp1(:,2)+2
This will work not matter how many items of
data are affected.
Matrix notation
Within a textbook, it is more normal to use
subscripts to represent the indices, so for
example:
 A1 ,1

A 2 ,1
A 
 A 3 ,1

 A 4 ,1
A 1, 2
A1 , 3
A 2,2
A2 , 3
A 3,2
A3 , 3
A 4,2
A4 , 3
A1 , 4 

A2 , 4

A3 , 4 

A4 , 4 
Often the
comma is
omitted for
low value
integers
B  B11
B12
B13 
Matrix notation
Matrix notation
Matrix notation
Matrix notation
Matrix notation
Flexibility
As an engineer you need to be flexible and
work out what notation a particular article,
website, book, etc is using.
Usually it will be obvious from the context
and usage as long as your understand the
core principles.
Which statement is false?
8

D  6

 2
1.
2.
3.
4.
5.
3
5
D12=-3
D21=6
D32=2
D23=2
D33=0
4
2

0

0 
Matrix dimensions
What are the dimensions of the following
matrices?
8

D  6

 2
3
5
4
2
A 
 1
C  2 1
5
2

0 ;

0 
3

E  4

 2
0
4
;
5
4
0
3
3
2
5
1
4
0
5 
 
B  6
 
 7 
4
0
4

5

3 
Matrix dimensions summary
• Dimensions are always number of rows
first and then number of columns second.
Typical notation is 2 by 3 or 4 by 1 or
similar.
Describe dimensions of matrices on
previous slide.
• Indexing begins from top left corner.
MATRIX TRANSPOSE
This is a very common operation so it is critical
that students are familiar with the notation.
There is nothing to understand as it is simply a
definition.
MATRIX TRANSPOSE
This is a very common operation so it is critical
that students are familiar with the notation.
There is nothing to understand as it is simply a
definition.
Matrix transpose definition
The transpose is determined by swapping
the position of all the indices, that is {1,2}
becomes {2,1}, etc.
T
Ai , j  A j ,i
You could view this a bit like doing a
reflection about the principle diagonal.
Diagonal elements do not change.
 2
A
 3
3
;
4
2
A 
3
T
 3
;
4 
Examples of Matrix transpose
Ai , j  A
 2
A   6

3
C  2 1
3
4 ;

4
5
T
j ,i
2
A  
3
T
4
0
4
Alternative insight.
The jth column of A is the jth
row of A transpose.
6
4
0 ;
3

4 
C
T
 21 


5


 4

 0 


 4 
 0 
Matrix transpose examples
Find the transpose of the following:
8

D  6

 2
3
3

E  4

 2
3
2
5
1
4
0
2

0

0 
5
4
5 
 
B  6
 
 7 
4

5

3 
Which is the correct transpose of G?
2
G  2

2
2 
 2 ;

 2
2
C 
2
2
1.
2.
3.
4.
5.
2
 2
;
 2
2
A
2
2
D 
2
A
B
C
D
None of these
2
2
2
2
 2
;
2 
2 

 2
 2
B 
 2
2
2
2

2
ADDITION AND SUBTRACTION
OF MATRICES
Addition and subtraction of
matrices
Add (or subtract) components in same
position, that is with the same {row,
column} index.
C  A B

C i , j  Ai , j  B i , j
Matrices must be same dimensions or you
cannot add them!
 2
A
 3
3
;
4
1
B
7
2 
;
 6
3
A B  
4
5 

 2
Do following matrices
computations
 2
A
 3
 2
A   6

3
3
;
4
3
4 ;

4
4
B
7
2 
;
 6

AB  

 2
B   12

8
0
6 ;

7

A  2B  









 2
G  
 3
1
C  
2
 3
F  
 1
1.
2.
3.
4.
5.
1
;
4
3
;
5
4
;
9
2
A 
0
2
D  
2
 2
;
8 
6
2 

 1
0
2
 0
E  
 2
G+C=F
G-C=-F
A+D=E
D-A=-E
None of these
2
2
8
 4

9 
Which is
incorrect
(could be
several)?
Do following matrix computations
 2
A
 3
2
A  1

3
3
;
4
3
5 ;

0
4
B
7
2 
;
 6
 2
B   1

 0
0
6 ;

4

A  3B  

T
AB
T




MATRIX handling in MATLAB
• Matrix
addition/subtraction
• Matrix multiplication
• Matrix powers
• Matrix inversion
• Extracting parts of
matrices
•
•
•
•
A+B, A-B, A-2*B+C
A*B
A^3
inv(A) or A^(-1)
• A([1,3],[2, 4])
MATRIX MULTIPLICATION
This is the most common skill you will need.
The technique is simply a definition that you need to
memorise.
There is no such thing as matrix division!
Basic rule for multiplication
(BY DEFINITION – not to understand)
Index {i,j} of result is determined from row {i}
of left hand matrix and column {j} of right
hand matrix.
Multiplying a row on left by a column on the
right is a ‘dot-product’ type of operation,
e.g.
e
a
b
c
 
 
f

d    ae  bf  cg  dh
g
 
h
What is the result of the
following ?
3
3
5
 2


6

 
 8 




1. 88
2. 52
3. 32
What is the result of the
following ?
3
1 1
1.
2.
3.
4.
2
2
 
3
 
7  4 
0 
 
2
18
21
27
Don’t know
What is the result of the
following ?
1
1.
2.
3.
4.
1 
 
2 0
 
 3 
1
6
NA
Not sure
KEY OBSERVATION
A row vector can only be multiplied onto a
column vector if the two vectors are the
same length.
Number of columns of left hand vector must
match the number of rows of the right
hand vector.
If vectors have different dimensions,
multiplication is not defined.
Rules for matrix multiplication
Assume we wish to do C=A*B.
A must have same number of columns as B
has rows
Columns
of A
Columns
of C
Rows
of C
C
A
=
Rows
of A
Columns
of B
B
Rows
of B
Basic rule for multiplication
Index {i,j} or result is determined from row {i} of left
hand matrix and column {j} of right hand matrix.
Multiplying a row on left by a column on the right is
a ‘dot-product’ type of operation, e.g.
a

k
a

c

 e
b
g
d 
 k

f 
b
c
n
m
h
i
n
m
e
 
d f
 ae  bf  cg  dh 
 





p g
 ke  nf  mg  ph 
 
h
 ag  bk
j 
   cg  dk
p
 eg  fk
ah  bn
ai  bm
ch  dn
ci  dm
eh  fn
ei  fm
aj  bp 

cj  dp

ej  fp 
Student problem
Solve the following multiplications.
2

4
2

4
5
6
5
6
1 7
6 

  4 5 
8
 6 2 
 1 7 9 1 
6 

   4 5 10 1 
8
 6 2 12 2 
Student problem 2 – compute C,D
2
A 
4
5
6
6
8
4
;
6
1 7


B  4 5 ;


 6 2 
C  AB ;
D  BA
KEY OBSERVATION
Two matrices can only be multiplied if the
column dimension of the left matrix
matches the row dimension of the right
The property of
matrix.
AB=BA is called
In general
commutativity
AB  BA
and does not
hold for matrices
in general.
In fact, it is possible that AB exists but
BA does not!
Uses of matrix multiplication
A simple example involves the change of coordinates
from one set to another – this is commonly needed
for robotics and mechanics.
X '  x cos   y sin 
V=(x,y)
y
Y’
X’
θ
x
Y '   x sin   y cos 
 X '  cos 
 
 Y '    sin 
sin    x 
 
cos    y 
SIMULTANEOUS EQUATIONS
Matrix/vector algebra is a compact and efficient method
for handling linear simultaneous equations and most
effective techniques deployed assume matrix
representation of this problem. [Details in semester 2]
MATLAB automatically handles simultaneous equations
with matrices and vectors which greatly simplifies
solution and manipulation. [Try this now!]
Using Matrices for simultaneous
equations
Consider a single linear equation and write
in MATRIX/VECTOR format, e.g.
3x  2 y  4
 x  5y  6
 3x  2 y   3


 x  5 y   1

x
[3 2 ]   4
 y

x
[  1 6 ]   6
 y
2  x  4
    
5   y  6 
Put following into matrix-vector
format
 5 x  6 y   1


x

3
y

2


x  z  2y  3 



3
z

6
y


4


 20 x  13 y  4 z  10 


 a  b  c  12 


b

2
c

3
d

4




 a  14 c  d  2 
 b  12 a  14 d  9 


Simultaneous equations
Assume user knows how to put simultaneous
equations into matrix vector format, ie.
AX=b
Can solve with MATLAB several different ways;
1. X=inv(A)*b
2. X=A\b
3. Numerically efficient code is discussed in text
books and semester 2 of ACS123.
1
Try on MATLAB
x  A b;
 1 


b 3 ;


  2 
2

A 1

 0
1
4
2
0 

2

8 
DETERMINANTS
You will cover solution of simultaneous equations in
semester 2. However, in order to prepare for this, you will
need familiarity and skill with the determinant of a matrix.
Final matrix topic here is define ‘determinant’.
NOTE: This is a definition and therefore must be learnt
as a fact.
ONLY DEFINED FOR SQUARE MATRICES
Determinant in brief
In conceptual terms, a determinant gives an
indication of the ‘magnitude’ of a matrix, that is, on
average, what scale of impact does it have on
vectors when multiplying those vectors.
You will notice that the effective magnification is
direction dependent, and so the determinant is an
average in some sense.
3

2
2  1  5 
     ;
3  1  5 
Scaled by factor of 5.
3

2
2  1   1 
     ;
3    1   1
Scaled by factor of 1.
Determinant – key fact
If the determinant is very small, then there exists a
least one direction, which when multiplied by the
matrix, results in a very small scaling factor.
10

18
1  1  11 
     ;
2  1  20 
Scaled by factor of 16
(approx).
10

18
1   0 . 1   0 . 1

   ;
2    0 .9   0 
Scaled by factor of 0.1
(approx).
A zero determinant means at least one direction
can be mapped to zero.
Determinant for 2 by 2 matrices
The determinant is defined as product of the
diagonal elements minus the product of the offdiagonal elements:
a
A
c
b
;
d
A  ad  bc
Note use of vertical lines around matrix is notation
use to define determinant.
Try this on the examples of the previous few
slides.
Determinant computations
3
A
2
10
B
18
2
;
3
1
;
2
A 
B 
Find the determinant of matrix D
1
D 
 5
1.
2.
3.
4.
5.
-12
28
12
-28
unsure
4

8
Find the determinant of matrix F
 1
F 
 4
1.
2.
3.
4.
5.
0
24
-24
-32
unsure
 3

12 
Determinant for 3 by 3 matrices
The determinant is defined using the concept of a
matrix of cofactors Aij. For convenience define the
original matrix with lower case and the matrix of
cofactors with upper case.
 a 11

A  a 21

 a 31
a 12
a 22
a 32
a 13 

a 23 ;

a 33 
Next, by definition:
 A11

cof ( A )  A 21

 A31
A12
A 22
A32
A13 

A 23

A33 
A  a 11 A11  a 12 A 12  a 13 A13
or  a 21 A 21  a 22 A 22  a 23 A 23
or  a 31 A31  a 32 A 32  a 33 A33
Defining the cofactor terms
A cofactor is taken as the determinant of a minor
with a sign change if appropriate to the position.
A minor is the matrix remaining when the row and
column associated to an element are crossed out.
 a 11

A  a 21

 a 31


sign  

 
a 12
a 22
a 32



a 13 

a 23 ;

a 33 


 ;

 
 a 21
min or ( a 13 )  
 a 31
a 22 

a 32 
A13  cof ( a13 )  a 21 a 32  a 22 a 31
Defining the cofactor terms
A cofactor is taken as the determinant of a minor
with a sign change if appropriate to the position.
A minor is the matrix remaining when the row and
column associated to an element are crossed out.
 a 11

A  a 21

 a 31


sign  

 
a 12
a 22
a 32





 ;

 
a 13 

a 23 ;

a 33 
 a 12
min or ( a 21 )  
 a 32
a 13 

a 33 
A21  cof ( a 21 )  [ a 12 a 33  a 32 a13 ]
Find the matrix of cofactors and
determinant
3

A 8

 0
6
1
1
5

0

2 
A  a 11 A11  a 12 A 12  a 13 A13
Find the matrix of cofactors and
determinant
 0

A 2

 2
3
1
1
4

2

5 
Determinant for 4 by 4 matrices
The determinant is defined using the concept of a
matrix of cofactors Aij (as for the 3 by 3 case)
For convenience define the original matrix with
lower case and the matrix of cofactors with upper
case.
Next, by definition:
A  a 11 A11  a 12 A 12  a 13 A13  a 14 A14
4
or 
a
ij
Aij
j 1
The extension to larger matrices should be obvious,
although this is not a paper and pen exercise.
Defining the cofactor terms for a 4 by 4
A cofactor is taken as the determinant of a minor
with a sign change if appropriate to the position.
A minor is the matrix remaining when the row and
column associated to an element are crossed out.
 a 1 ,1

a 2 ,1

A
 a 3 ,1

 a 4 ,1
a 1, 2
a1, 3
a 2,2
a 2 ,3
a 3,2
a 3,3
a 4,2
a 4 ,3
a1, 4 

a 2,4

a 3,4 

a 4,4 
Remarks on 4 by 4 determinants
A minor is now a 3 by 3 matrix and therefore
each cofactor requires the computation of
the determinant of a 3 by 3
For a 5 by 5 matrix, the minors would each
require the computation of a 4 by 4
determinant.
Determinants of larger matrices are best done using
properties and tricks to minimise the numeric
computations. Alternatively, and more likely, use a
computer.
Properties of determinants
1. Adding a multiple of any row to another row does not
change the determinant.
2. If two rows (or two columns) are equal then the
determinant is zero.
3. Scaling any row by λ results in a scaling of the
determinant by λ
4. Interchanging rows (or columns) changes the sign of the
determinant.
5. A lower or upper triangular matrix has a determinant
given as product of diagonal terms.
6. The determinant of a matrix and determinant of its
transpose are equal
Properties of determinants
1.
Adding a multiple of any row to another row does not change the
determinant.
Properties of determinants
2. If two rows (or two columns) are equal then the determinant is zero.
Properties of determinants
3. Scaling any row by λ results in a scaling of the determinant by λ.
Properties of determinants
4. Interchanging rows (or columns) changes the sign of the
determinant.
Properties of determinants
5. A lower or upper triangular matrix has a determinant given as product of
diagonal terms.
Properties of determinants
6. The determinant of a matrix and determinant of its transpose are equal
Find the determinant (using properties)
3

A 0

 0
6
1
0
5

0

2 
Check answers using MATLAB – e.g det(A)
Find the determinant (using properties)
 1

22

B 
 5

104
0
0
2
0
18
3
56
13
0

0

0

2
Check answers using MATLAB – e.g det(A)
Find the determinant (using properties)
1

A 2

 1
6
1
4
5

0

2 
Find the determinant (using properties)
 1

3
B  
 5

104
4
5
 12
 15
18
3
56
13
0

0

7

2
Find the determinant (using properties)
1

A 6

 4
 1

3
B  
 5

104
6
1
0
5

0

0 
2
0
 12
5
0
0
56
13
0

0

0

2
Adjoint
The adjoint matrix is the transpose of the
matrix of cofactors.
This will be used in semester 2 for solving
linear simultaneous equations.
All the cofactors need to be computed! THIS
IS TEDIOUS!
However: The adjoint is not needed if only
the determinant is required.