Matrix
Definition: A matrix is a rectangular array of
numbers enclosed by a pair of bracket and is
denoted by a capital letters A, B etc.
2 3 7
A

1

1
5


1 3 1 
B  2 1 4
4 7 6
Both A and B are examples of matrix.
1
Matrix
In the matrix
 a11
a
A   21


 am1
a12
a22
am 2
a1n 
a2 n 


amn 
= [aij]
numbers aij (i =1,2,--- m , j=1,2,---n)are called
elements. First subscript indicates the row;
second subscript indicates the column. The
matrix consists of mn elements
number of rows(m) by number of columns(n) of
a matrix is called order of the matrix and is
written as m  n (read m by n)
2
Matrix
Types of matrices
Square matrix:
When m = n, i.e.,
 a11
a
A   21


 an1
a12
a22
an 2
a1n 
a2 n 


ann 
A is called a “square matrix of order n x n
elements a11, a22, a33,…, ann called diagonal
elements. The line along which they lie is
called the principal diagonal.
n
  aii
i 1
 a11  a22
 ... 
ann
is called the trace of A.
3
Matrix
Types of matrices
Column matrix:
A matrix with only one column is
called Column Matrix , i,e
1 
A  0 
 
8
Matrix A is a column matrix of
Order 3x1
4
Matrix
Types of matrices
Row matrix:
A matrix with only one row is
called row matrix , i,e
A  1 2 3 4 5
Matrix A is a row matrix of
Order 1x5
5
Matrix
Types of 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.
6
Matrix
Types of matrices
Equal matrices
Example:
 1 0
A


4
2


and
a b 
B

c
d


Given that A = B, find a, b, c and d.
if A = B, then a = 1, b = 0, c = -4 and d = 2.
7
Matrix
Types of matrices
Zero matrix or Null Matrix:
Every element of a matrix is zero, it is called
a zero matrix, i.e.,
0 0
0 0
A


0 0
0
0 


0
8
Matrix
Types of matrices
A square matrix whose elements aij = 0, for
i > j is called upper triangular, i.e.,  a11 a12
0



0
a22
0
A square matrix whose elements aij = 0, for
i < j is called lower triangular, i.e.,  a11 0
a
 21


 an1
a22
an 2
a1n 
a2 n 


ann 
0
0 


ann 
9
Matrix
Types of matrices
Diagonal matrix:
A square matrix whose elements aij = 0, for i  j
is called diagonal matrix, i.e.,
 a11
0
D


0
0
a22
0
and is denoted by
0 
0 


ann 
D  diag[a11 , a22 ,..., ann ]
10
Matrix
Types of matrices
Scalar matrix:
A square diagonal matrix whose diagonal
elements are equal but not unity is called scalar
matrix, i.e.,
11
Matrix
Types of matrices
Identity matrix:
A square matrix whose elements aij = 0, for
i  j and aij = 1, for i = j is called identity
matrix or unit matrx and is denoted by I
1 0 0 
Examples of identity matrices: 1 0 and 0 1 0


0 1 
0 0 1 
Properties: AI = IA = A
12
Matrix
Types of matrices
Transpose matrix:
The matrix obtained by interchanging the
rows and columns of a matrix A is called the
transpose of A (write AT or A/ ).
For a matrix A = [aij], its transpose AT = [bij],
where bij = aji.
Example:
1 2 3
A

4
5
6


The transpose of A is
1 4 
AT   2 5 
 3 6 
13
Matrix
Types of matrices
Transpose matrix:
Properties
(AT)T = A and (lA)T = l AT
(A + B)T = AT + BT
(A - B)T = AT - BT
(AB)T = BT AT
14
Matrix
Types of matrices
Symmetric matrix:
A square matrix A whose elements aji = aij for
all i and j is called symmetric.
Example:
1 2 3 
A   2 4 5
 3 5 6 
is symmetric.
Properties: AT = A
15
Matrix
Types of matrices
Skew-symmetric matrix:
A square matrix A whose elements aji = - aij for
i  j and aij  0 for i  j is called skew-symmetric.
2  1
0
Example: A   2 0 2  is skew-symmetric.
 1  2 0 
Properties: AT = - A
16
Matrix
Types of matrices
Conjugate of a matrix:
17
Matrix
Types of matrices
18
Matrix
Types of matrices
Conjugate of a matrix:
19
Matrix
Types of matrices
20
Matrix
Types of matrices
Hermitian matrices:
21
Matrix
Types of matrices
Skew-Hermitian matrices:
22
Matrix
Types of matrices
Sub-matrices of a matrix (square or
rectangular:
23
Matrix
Types of matrices
Principal submatrices of a matrix:
24
Matrix
Types of matrices
Leading submatrices of a matrix:
25
Matrix
Types of matrices
Determinant of a square matrix:
26
Matrix
Determinants
Determinant of order n
Determinant of n n matrix
 a11
a
A   21


 an1
a12
a22
an 2
a1n 
a2 n 


ann 
is denoted | A | and is representation by
27
Matrix
Determinants
whose order is n
28
Matrix
Determinants
The following properties are true for
determinants of any order.
1. If every element of a row (column) is zero,
e.g.,
1 2
0 0
 1 0  2  0  0 ,
2. |AT| = |A|
then |A| = 0.
determinant of a matrix
= that of its transpose
3. |AB| = |A||B|
29
Matrix
Types of matrices
Minor of the matrix:
30
Matrix
Algebra of matrices
Addition and subtraction of matrices
Two matrices of the same order are said to
be conformable for addition or subtraction.
Two matrices of different orders cannot be
added or subtracted, e.g.,
2 3 7
1 1 5 


1 3 1 
2 1 4


 4 7 6 
are NOT conformable for addition or
subtraction.
31
Matrix
Algebra of matrices
Addition and subtraction 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.
1 2 3 
 2 3 0
Example: if A  0 1 4 and B   1 2 5


Evaluate A + B and A – B.


2  3 3  0   3 5 3
 1 2
A B  



0

(

1)
1

2
4

5

1
3
9

 

2  3 3  0   1 1 3 
 1 2
A B  



0

(

1)
1

2
4

5
1

1

1

 

32
Matrix
Algebra of matrices
Scalar multiplication
Let l be any scalar and A = [aij] is an m  n
matrix. Then lA = [laij] for 1  i  m, 1  j  n,
i.e., each element in A is multiplied by l.
Example:
1 2 3 
A
.
0
1
4


Evaluate 3A.
 3  1 3  2 3  3  3 6 9 
3A  



3

0
3

1
3

4
0
3
12

 

In particular, l  1, i.e., A = [aij]. It’s called
the negative of A. Note: A  A = 0 is a zero matrix
33
Matrix
Algebra of matrices
Properties
Matrices A, B and C are conformable,
A + B = B + A
(commutative law)
A + (B +C) = (A + B) +C
(associative law)
l(A + B) = lA + lB, where l is a scalar
(distributive law)
Can you prove them?
34
Matrix
Algebra of matrices
Matrix multiplication
Two matrices may be multiplied if the number
of columns in left product must equal the
number of rows in right product. This condition
is called conformable for 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
matrix C = AB, where C= [cij] with
p
cij   aik bkj  ai1b1 j  ai 2b2 j  ...  aip bpj
k 1
for 1  i  m, 1  j  n.
35
Matrix
Algebra of matrices
Matrix multiplication
Example:
 a11 a12
A  a21 a22
a31 a32
 a11 a12
Solution : AB  a21 a22
a31 a31
a13 
a23 
a33 
a13 
b11 b12 b13 
a23  , B  b21 b22 b23 


a33 
b31 b32 b33 
, find AB.
b11 b12 b13 
b b

b
 21 22 23 
b31 b31 b33 
 a11b11  a12b21  a13b31 a11b12  a12b22  a13b32
 a21b11  a22b21  a23b31 a21b12  a22b22  a23b32
 a11b31  a12b31  a33b31 a31b12  a32b22  a33b32
a11b13  a12b23  a13b33 
a21b13  a22b23  a23b33 
a31b13  a32b23  a33b33 
36
Matrix
Algebra of matrices
Matrix multiplication
Example:
1 2 3 
A

0
1
4


,
 1 2 
B   2 3 ,
 5 0 
Evaluate C = AB.
Answer: Since order of A is 2  3 and order
of B is 3 2 , i,e conformable for multiplication
AB
 1 2
1 2 3  

2
3
0 1 4  



 5 0 
c21  0  (1)  1 2  4  5  22
cont
37
Matrix
Algebra of matrices
Matrix multiplication
 c11  1  (1)  2  2  3  5  18
 1 2 
1 2 3  
 c12  1  2  2  3  3  0  8

0 1 4  2 3  c  0  (1)  1  2  4  5  22


 21

5
0

  c  0  2  1 3  4  0  3
 22
 1 2
1 2 3  
18 8

C  AB  
2 3  



0
1
4
22
3





5
0


38
Matrix
Algebra of matrices
Matrix multiplication
In particular, A is a 1  m matrix and
 b11 
B is a m  1 matrix, i.e.,
A   a11 a12 ... a1m 
b 
B   21 
 
 
bm1 
then C = AB is a scalar.
m
C   a1k bk1  a11b11  a12b21  ...  a1mbm1
k 1
39
Matrix
Algebra of matrices
Matrix multiplication
BUT BA is a m  m matrix!
 b11 
 b11a11 b11a12
b 
b a
b21a12
21 
21 11


BA 
a11 a12 ... a1m  

 

 

bm1 
bm1a11 bm1a12
b11a1m 
b21a1m 


bm1a1m 
So AB  BA in general !
40
Matrix
Algebra of matrices
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
AB = AC NOT necessarily imply B = C
41
Matrix
Algebra of matrices
However, if two square matrices A and B
such that AB = BA, then A and B are said to
be commute.
If A and B such that AB = -BA, then A and B
are said to be anti-commute.
42
Matrix
Algebra of matrices
43
Matrix
Algebra of matrices
44
Matrix
Orthogonal matrix
A matrix A is called orthogonal if AAT = ATA = I,
i.e., AT = A-1
Example: prove that
orthogonal.
Since,
 1/ 3

T
A   1/ 6

 1/ 2
1/ 3 1/ 6

A  1/ 3 2 / 6

1/ 3 1/ 6
1/ 3 

2 / 6 1/ 6 

0
1/ 2 
1/ 3
1/ 2 

0 

1/ 2 
is
. Hence, AAT = ATA = I.
Can you show the
details?
We’ll see that orthogonal matrix represents a
rotation in fact!
45
Matrix
Algebra of matrices
(AB)-1 = B-1A-1
(AT)T = A and (lA)T = l AT
(A + B)T = AT + BT
(AB)T = BT AT
46
Matrix
Algebra of matrices
Problem: If A is a square matrix , then
show that A + AT is symmetric and A – AT
is skew-symmetric.
Proof: Since A and B are symmetric ,i,e.
A = AT and B =BT . (A + AT )T = AT +(AT) T =
AT +A = A + AT . Hence A+B is symmetric .
Again (A - AT )T = AT -(AT) T = AT –A =
-(A - AT ). Hence A – AT is skew-symmetric.
47
Matrix
Algebra of matrices
Problem: If A and B are symmetric (skewsymmetric ) matrices , then A +B and A – B
are also symmetric.
Proof: Since A and B are symmetric , i,e
A = AT and B =BT . (A + B )T = AT +BT = A +B .
Hence A+B is symmetric . Again (A - B )T =
AT -BT = A -B . Hence A – B is symmetric.
48
Matrix
Algebra of matrices
Problem: Every square matrix can be
expressed in one and only one way as the
sum of a symmetric and a skew-symmetric
matrix.
Solution: For a square matrix A, we have
A + AT is symmetric and A – AT is skewsymmetric. Obviously
(A + AT ) = P is also
symmetric and ( A – AT ) = Q is also
skew-symmetric.
(Cont)
49
Matrix
Algebra of matrices
Clearly A = (A + AT ) + ( A – AT ) = P +Q
which is the sum of symmetric and skewsymmetric.
Now we are show that this representation
is unique
(Cont)
50
Matrix
Algebra of matrices
Let A=P+Q and A=R+S be two
representation of A with P , R symmetric
and Q , S skew-symmetric
Now from A=R+S , we have = AT =(R+S) T
= RT +S T = R –S
So, P = (A + AT ) = (R+S+R-S)=R
Q = (A -AT ) = (R+S – R +S)= S
Hence the representation is unique
51
Matrix
Algebra of matrices
Problem: If A and B are symmetric
matrices , then show that AB +BA is
symmetric and AB – BA is skew-symmetric.
Proof: Since A and B are symmetric , i,e
A = AT and B =BT .
(AB + B A)T = (AB)T +(BA)T = BT AT +AT BT
= B A +A B= A B+ B A . Hence AB+BA is
symmetric .
(Cont)
52
Matrix
Algebra of matrices
Again (AB - B A)T = (AB)T -(BA)T = BT AT - AT BT
= B A -A B= - (A B- B A) . Hence AB - BA is
skew-symmetric .
53
Matrix
Determinants
Cofactor of elements
The cofactor of an element aij is obtained
by multiplying the minor of element aij by
(-1)i+j and is denoted by Aij
1 0 1 
Example A  2 1 2 , cofactor of element a12  0
3 1 0
is
1 2
A12  (1)
2 2
3 0
6
54
Matrix
Determinants
Cofactor of elements
Properties

3
j 1
 A , when i  j
aij Aij  
o , when i  j
1 0 1 
Proof: Let A  2 1 2
3 1 0
(Cont)
55
Matrix
Determinants
•Cofactor of elements
A11 
1 2
1 0
A21  
A31 
 2, A12  
0 1
1 0
0 1
1 2
 1, A22 
1 1
3 0
 1, A32  
2 2
3 0
 6, A13 
 3, A23  
1 1
2 2
 0, A33 
(Cont)
1 0
3 1
1 0
2 1
2 1
3 1
 1
 1
1
56
Matrix
Determinants
•Cofactor of elements(cont)
1 0 1
A  2 1 2  1(0  2)  0(0  6)  1(2  3)   3
3 1 0
A  a11 A11  a12 A12  a13 A13  1(2 )  0( 6 )  (1)  3
A  a21 A11  a22 A12  a23 A13  2(2 )  1( 6 )  2(1)  0
proof
57
Matrix
Determinants
•Cofactor of elements
Example: Find the cofactor matrix of the
matrix
Solution:
A11 
1 2
1 0
1 0 1 
A  2 1 2
3 1 0
 2, A12  
2 2
3 0
(Cont)
 6, A13 
2 1
3 1
 1
58
Matrix
Determinants
•Cofactor of elements(cont)
0 1
1 1
1 0
A21  
 1, A22 
 3, A23  
 1
1 0
3 0
3 1
A31 
0 1
1 2
 1, A32  
1 1
2 2
 0, A33 
1 0
2 1
1
59
Matrix
Determinants
For any 2x2 matrix
 a11
A
 a21
a12 
a22 
Its inverse can be written as
Example: Find the inverse of
The determinant of A is -2
Hence, the inverse of A is
1
A 
A
1
 a22
 a
 21
a12 
a11 
 1 0 
A

1
2


0 
 1
A 

1/ 2 1/ 2 
1
How to find an inverse for a 3x3 matrix?
60
Determinants of order 3
Consider an example:
1 2 3
A   4 5 6 
7 8 9 
Its determinant can be obtained by:
1 2 3
4 5
1 2
1 2
A  4 5 6 3
6
9
7 8
7 8
4 5
7 8 9
 3  3  6  6  9  3  0
You are encouraged to find the determinant
by using other rows or columns
61
Matrix
Adjoint of a square matrix
The adjoint of a square matrix A is the
transpose of the matrix formed by the
cofactors of the elements of the
determinant of the matrix A and is denoted
by adjA
62
Matrix
Adjoint of a square matrix
Properties
* A(adjA)  (adjA) A  A I
* adjAB  (adjB) (adjA)
* (adjA)  adj ( A )
T
T
63
Matrix
Inverse matrix
If two non singular (i,e A  0 ) square
matrices A and B such that AB = BA = I, then
B is called the inverse of and is denoted by
the symbol A-1 and is defined by
adjA
A 
A
1
Properties
1 T
T 1
* (A )  (A )
1
1
1
* ( AB)  B A
(Cont)
64
Matrix
Inverse matrix
Properties
Pr oof : ( AB) ( AB)
1
I
 B 1 A1 ( AB) ( AB) 1  B 1 A1 I
1
1
1
1
1
 B (( A A) B ) ( AB)  B ( A I )
1
1
1
 B ( IB ) ( AB)  B A
1
 ( B 1 B ) ( AB) 1  B 1 A1
1
1
1
1
1
 I ( AB)  B A  ( AB)  B A
65
1
Matrix
Adjoint of a square matrix(cont)
Example: Find the adjoint matrix of the
matrix
1 2 3


A  2 3 1
4 3 3
Solution:
3 1
2 1
2 3
A11 
 6, A12  
 2, A13 
 6
3 3
4 3
4 3
(Cont)
66
Matrix
Adjoint of a square matrix(cont)
A21  
A31 
0 1
1 0
0 1
1 2
 1, A22 
1 1
3 0
 1, A32  
 3, A23  
1 1
2 2
 0, A33 
1 0
3 1
1 0
2 1
 1
1
1  1
6


 adjA   2  3 0  Ans.
 6  1 1 
67
Matrix
Inverse matrix
Find Cofactor matrix
1 2 3 
of A  0 4 5
1 0 6 
and find A-1
Answer: Here cofactor are
A11 
A21 
A31 
4 5
0 6
2 3
0 6
2 3
4 5
 24
 12
 2
A12  
A22 
0 5
1 6
1 3
1 6
A32  
5
3
1 3
0 5
 5
(Cont)
A13 
0 4
1 0
A23  
A33 
 4
1 2
1 0
1 2
0 4
2
4
68
Matrix
Inverse matrix
Hence Cofactor matrix of
given by:
1 2 3 
A  0 4 5 
1 0 6 
is then
 24 5 4 
 12 3 2 


 2 5 4 
(Cont)
69
Matrix
Inverse matrix
Inverse matrix of
1 2 3 
A  0 4 5 
1 0 6 
is given by:
 24 5 4 
 24 12 2 
1 
1 


1
A 

12
3
2

5
3

5


A
22 
 2 5 4 
 4 2
4 
T
 12 11  6 11 1 11 
  5 22 3 22 5 22 
  2 11 1 11
2 11 
70