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Lecture 1
1
Matrices and Determinants

Matrix ‫المصفوفة‬
‫المرتبة‬ ‫األرقام‬ ‫من‬ ‫مجموعة‬ ‫هي‬ ‫لمصفوفات‬
.
‫وتعد‬ ‫جدول‬ ‫داخل‬ ‫وأعمدة‬ ‫صفوف‬ ‫في‬
‫والفلك‬ ‫الفيزياء‬ ‫في‬ ‫قوية‬ ‫أداة‬ ‫المصفوفات‬
‫تستخدم‬ ‫حيث‬ ،‫واالقتصاد‬ ‫الميكانيكي‬ ‫والطب‬
‫المختلفة‬ ‫المفاهيم‬ ‫من‬ ‫العديد‬ ‫عن‬ ‫للتعبير‬
‫وتحليل‬ ‫تمثيل‬ ‫في‬ ‫المصفوفات‬ ‫تساعد‬
،‫وفعالة‬ ‫منظمة‬ ‫بطريقة‬ ‫المختلفة‬ ‫البيانات‬
‫في‬ ‫والمتخصصين‬ ‫والمهندسين‬ ‫للباحثين‬ ‫وتتيح‬

1. Matrices
2. Operations of matrices
3. Types of matrices
4. Properties of matrices
5. Determinants
6. Inverse of a 33 matrix
3

2 3
7
4
A 


1 1
5
1.1 Matrices
4

1 3
1
B 

2
1


4
6
Both A and B are examples of matrix. A matrix
is a rectangular array of numbers enclosed by a
pair of bracket.
Why matrix?

How about solving
5

 x  y 
7,
3x  y 
5.






x  y  2z  7,
2x  y  4z  2,

5x  4 y  10z  1,
3x  y  6z  5.
Consider the following set of equations:
It is easy to show that x = 3 and
y = 4.
Matrices can help…
1.1 Matrices

21



 m1 m2
mn 
6
a1n

2n


 a11 a12
 a
22
a a
A 
 
a
a a
In the matrix
numbers aij are called elements. First subscript
indicates the row; second subscript indicates
the column. The matrix consists of mn elements
It is called “the m  n matrix A = [aij]” or simply
“the matrix A ” if number of rows and columns
are understood.
1.1 Matrices

6
Square matrices
When m = n, i.e.,
is called the trace of A.
21



n
2
 n1
nn 
a1n

2n


a11 a12
a
22
a a
A 
 
a
a a
n
i1
A is called a “square matrix of order n”
or “n-square matrix”
elements a11, a22, a33,…, ann called diagonal
elements.
  aii
 a11  a22  ...  ann
1.1 Matrices

Equal matrices
Two matrices A = [aij] and B = [bij] are said to
be equal (A = B) iff each element of A is equal
to the corresponding element of B, i.e., aij = bij
for 1  i  m, 1  j  n.
iff pronouns “if and only if”
if A = B, it implies aij = bij for 1  i  m, 1  j  n;
if aij = bij for 1  i  m, 1  j  n, it implies A = B.
8
1.1 Matrices

Equal matrices
9
1.1 Matrices
1 0
A
 4
 

a b

B 
2 c d

  

Example: and
Given that A = B, find a, b, c and d.
if A = B, then a = 1, b = 0, c = -4 and d =
2.

Zero matrices
Every element of a matrix is zero, it is called
a zero matrix, i.e.,
0
0 0
0
0 0
0





0
0

 
A 

1
0
1.1 Matrices

10
1.2 Operations of matrices
Sums of matrices
If A = [aij] and B = [bij] are m  n matrices,
then A + B is defined as a matrix C = A + B,
where C= [cij], cij = aij + bij for 1  i  m, 1  j  n.
2
3
1

0
4



5

3
0
1 2
 
Example: if A 
1
and B 

2
Evaluate A + B and A – B.
9

0  (1) 4  5 1
 

A  B 
 1  2

3

3  0

1

0  (1) 4  5
2  3 3  0

 3 5
3
1  2 3
1
  
A  B 
 1  2

11
Sums of matrices
Two matrices of the same order are said to
be conformable ‫متأقلمة‬for addition or
subtraction.
Two matrices of different orders cannot be
added or subtracted, e.g.,
are NOT conformable for addition or
subtraction.
2 3
7
1 1
5


1 3
1
1
7

2
4




4

6



13
Scalar multiplication
Let  be any scalar and A = [aij] is an m  n
matrix. Then A = [aij] for 1  i  m, 1  j  n,
i.e., each element in A is multiplied by .

0
4

 
A 
1
Example: 3
. Evaluate
3A.

0
3 
0
3 
4
 

2
1
3  2 3  3

3 6
9 
3 1 3
12

3A 
 3
1
In particular,   , i.e., A = [aij]. It’s called
the negative of A. Note:   A = 0 is a zero

14
Properties
Matrices A, B and C are conformable,
A + B = B + A
A + (B +C) = (A + B) +C
(commutative law)
(associative law)
(A + B) = A + B, where  is a scalar
(distributive law)
Can you prove them?

Properties
Example: Prove (A + B) = A +
B. Let C = A + B, so cij = aij + bij.
Consider cij =  (aij + bij ) = aij +
bij, we have,
C = A + B.
Since C = (A + B), so (A + B) =
A + B 15

15
Matrix multiplication
If A = [aij] is a m  p matrix and B = [bij] is a
p  n matrix, then AB is defined as a m  n
ij
matrix C = AB, where C= [c ] with
p
k
1
cij   aik bkj  ai1b1 j  ai 2b2 j  ...  aipbpj
1
2
3

0
4


1
2 3



 5
0
Example: A 
1
, B  
2
and C = AB.
Evaluate c21.
 1
2
1 2 3
 2
3


0
4
 

0

21
c  0 (1) 1 2  4  5  22
for 1  i  m, 1  j  n.

16
1 2
3
0 1
4
1
2


 5
0
Example: A 

 , B 
 2
3
, Evaluate C =
AB.
21
c22
 c11  1 (1)  2  2  3  5  18
1
2
 c  1 2  2  3  3  0  8
1 2
3
1

2
3 


c

0
4

12
 0  (1)  1 2  4  5  22
 

5

0


 0  2  1 3  4  0  3



1
2
C  AB 
1
2 3  2 3 
18
8

0
2
2
3

4
  


0


In particular, A is a 1  m matrix and
B is a m  1 matrix, i.e.,
m
11 1m
A 
a
a12 ...
a
21



 b11

18
b

 B 

b

m1

then C = AB is a scalar. C   a1k bk1  a11b11  a12b21  ...  a1mbm1
k 1

BUT BA is a m  m matrix!

21 21 11 21 12
11 12

 





21 1m


19
1m

m1


m1 11
m1 12
m1 1m

 b11

 b11a11 b11a12
b11a1m


b

b a b
a
b a
BA 

a a ... a
b


b a b
a
b a
So AB  BA in general !

Properties
Matrices A, B and C are
conformable,
A(B + C) = AB + AC
(A + B)C = AC + BC
A(BC) = (AB) C
AB  BA in general
AB = 0 NOT necessarily imply A = 0
or B = 0 20

n n
21
n n n
k 1 k
1
k 1 k 1 k
1
 ckj )
yij   aik xkj   aik
(bkj
 (aik bkj  aik ckj )   aik bkj   aik ckj
Properties
Example: Prove A(B + C) = AB + AC where A,
B
and C are n-square matrices
Let X = B + C, so xij = bij + cij. Let Y = AX, then
So Y = AB + AC; therefore, A(B + C) = AB + AC

1.3 Types of matrices
22
Identity matrix
The inverse of a matrix
The transpose of a matrix
Symmetric matrix
Orthogonal matrix

0

0




0

 nn

i > j is called upper triangular, i.e.,
a11
a12
22
a1n

2n


a a
a
A square matrix whose elements aij = 0, for
11
21 22
0 0

0







n1
23
n
2
nn

i < j is called lower triangular, i.e.,
a

a
a

a
a a
Identity matrix
A square matrix whose elements aij = 0, for

i  j , i.e., 11
22
0


0
0





0

 nn

24
a 0
a
D 

0 a
Identity matrix
Both upper and lower triangular, i.e., aij = 0,
for
is called a diagonal matrix, simply
D  diag[a11, a22 ,..., ann ]

In particular, a11 = a22 = … = ann = 1, the
matrix is called identity matrix.
Properties: AI = IA = A
25
1
0
Examples of identity matrices:

0
1
1
0
0
 and 0 1
0
0




0

1


Identity matrix

If A and B such that AB = -BA, then A and B
are said to be anti-commute.
26
Special square matrix
AB BA in general. However, if two square
matrices A and B such that AB = BA, then A
and B are said to be commute.
Can you suggest two matrices that must
commute with a square matrix A?
Ans:
A
itself,
the
iden
ti
ty
mat
rix
,
..

26
The inverse of a matrix
If matrices A and B such that AB = BA = I,
then B is called the inverse of A (symbol: A-1);
and A is called the inverse of B (symbol: B-1).
0

 6 2
3
B  1 1

1 0

1


Show B is the the inverse of matrix A.
1 2
3
3
2
3




1

4


Example: A 
1
1 0
0
1
0





1


Ans: Note that AB  BA  0
Can you show the
details?

The transpose of a matrix
The matrix obtained by interchanging the
rows and columns of a matrix A is called the
transpose of A (write AT).
28
2
3
5

4
6



Example: A 
1 1
4 5


2


3
6
The transpose of A is AT
For a matrix A = [aij], its transpose AT = [bij],
where bij = aji.

Symmetric matrix
A matrix A such that AT = A is called symmetric,
i.e., aji = aij for all i and j.
A + AT must be symmetric. Why?
29
is symmetric.
1 2 3

4
5
5



3

6


Example: A 
2
A matrix A such that AT = -A is called
skew- symmetric, i.e., aji = -aij for all i and j.
A - AT must be skew-symmetric. Why?

Orthogonal matrix
A matrix A is called orthogonal if AAT = ATA =
I, i.e., AT = A-1
is
orthogonal.
1/
3
2


1/ 6 1/
3 2 / 6 0


 

1
/
3 1/ 6 1/ 2

Example: prove that A 
1/
0 1/
Since, AT
3


 1/ 3 1/ 3 1/

  1/ 6 2 / 6
1/
6  . Hence, AAT = ATA =
I.
2


1/
2




Can you show the
details?
We’ll see that orthogonal matrix
represents a rotation in fact! 30

(AB)-1 = B-1A-1
(AT)T = A and (A)T =  AT
(A + B)T = AT + BT
(AB)T = BT AT
31
1.4 Properties of matrix

1.4 Properties of matrix
32
Example: Prove (AB)-1 = B-1A-1.
Since (AB) (B-1A-1) = A(B B-1)A-1 = I and
(B-1A-1) (AB) = B-1(A-1 A)B = I.
Therefore, B-1A-1 is the inverse of matrix AB.

1.5 Determinants
Consider a 2  2 matrix: a

a

A 
a11 a12

 21
22

Determinant of order 2
21
33
a12
22
a
a
| A |
a11
 a11a22  a12 a21
 Determinant of A, denoted | A |, is a
number and can be evaluated by

21
a12
22
a
a
| A |
a11
 a11a22  a12 a21
Determinant of order 2
easy to remember (for order 2 only)..
2
3 4
Example: Evaluate the determinant:
1
1
3
4
34
2
 1 4  2  3  2
1.5 Determinants
+
-

1.5 Determinants
2. |AT| = |A|
3. |AB| = |A||B|
determinant of a matrix
= that of its transpose
The following properties are true for
determinants of any order.
1. If every element of a row (column) is zero,
e.g., 1 2
 1 0  2  0  0 , then |A| = 0.
0 0
35

Example: Show that the determinant of any
orthogonal matrix is either +1 or –1.
For any orthogonal matrix, A AT = I.
Since |AAT| = |A||AT | = 1 and |AT| = |A|, so |A|2 = 1
or
|A| = 1.
36
1.5 Determinants

1.5 Determinants
For any 2x2 matrix
a

a

A 
a11 a12

 21
22

Its inverse can be written as
a
37
A1

1 
a22

A a
a12 
 21 11

0

Example: Find the inverse of A 
1

1
2

 
The determinant of A is -2
Hence, the inverse of A is A1

 1 0

1/
2
1/
2
 
How to find an inverse for a 3x3 matrix?

1.5 Determinants of order 3
1 2
3
5
8
6

Consider an example: A 
4



7

9


Its determinant can be obtained by:
1 2 3
A  4 5 6  3
4
7 8 9
5
 6
1 2
 9
1
2
7 8 7 8 4 5
 33  6 6  9 3  0
You are encouraged to find the determinant
by using other rows or columns 38

1.6 Inverse of a 33 matrix
Cofactor matrix of 5

1 2
3
A  0
4
0



1

6


The cofactor for each element of matrix A:
11
0 6
A 12
1 6
13
1 0

4

0
5
 24 A  
0 5
 5 A
4
 4
21
0 6
A 
2 22
1 6
3
 12
A

1 3
 3 23
1 0
A  
1 2
 2
31
4 5
A 
2 3
 2 32
0 5
A  
1 33
39
0 4
3
 5
A

1 2
 4

is then given
40
Cofactor matrix of
by:
5

1 2
3
A  0
4
0



1

6


2

 24 5 4
12 3

 2 5

4


Inverse matrix of is given by:
5

1 2
3
A  0
4
0



1

6


2


1 
5
24 12 2
3
5
2
A

22

 24 5
4
T
A1

1 12 3
5

2

4



4

4


12
11
41
  5
22
 6 11 1 11
 3 22 5
22


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