Module 2.0 Matrices

MATHEMATICS UNIT : FET/FIB
MAT 1064

2.1 DEFINITION & TYPES OF MATRICES
Definition
 A matrix is a rectangular array of numbers enclosed between brackets.
 The general form of a matrix with m rows and n columns is

 a11 a12 a13  a1n 
 
 a21 a22 a23  a2n 

 a31 a32 a33  a3n  m rows
      
 
am1 am2 am3  amn 

n columns

 The order or dimension of a matrix of m rows and n columns is m x n.

 The individual numbers that makes up a matrix are called its entries or elements, aij and they
are specified by their row and column position.

 The matrix for which the entry is in ith row and jth column is denoted by [ aij ].

Example 1

Let = [ 5 6 1
−2 3
2] ; (a) what is the order of matrix A? , (b) Identify the values of a21 and a13 .
−7

(a) Since A has 2 rows and 3 columns, the order of A is 2 x 3.

(b) The entry a21 is in the second row and the first column. Thus, a21 = -2.

The entry a13 is in the first row and the third column, so a13 = 1 .
2

Example 2

Given = [ ]3×3. Find matrix A if = {2 ;+ ≤
; >

a11 = 1(1) = 1 a12 = 1(2) = 2 a13 = 1(3) = 3

a21 = 2(1) + 2 = 4 a22  2(2) = 4 a23 = 2(3) = 6

a31 = 2(1) + 3 = 5 a32 = 2(2) + 3 = 7 a33 = 3(3) = 9

USER 1

MATHEMATICS UNIT : FET/FIB
MAT 1064

Equality of Matrices

Two matrices are equal if they have the same dimension and their entries are equal.

Example 3

Let A  3  a 6 4 and B   9 6c 4
 8 4b 2 , 2  3d 8 2 .


If = , find the values of , , and .

3 – = 9 4 = −8 6 – = 6 2 – 3 = 8

= −6 = −2 = 0 = −2

Types of Matrices

1. Row Matrix is a (1 x n) matrix (one row);

 A  a11 a12 a13  a1n

Example 4 B  1 0 7 8 4 3 5

A  1 2 3 4 5;

2. Column Matrix is a (m x 1) matrix (one column);

 a11  Example 5
 
 a21  2

A   a31  A  0 , B  3
   4 5
 
am1  7

3. Square Matrix is a n x n matrix which has the same number of rows as columns.

Example 6

A  1 3 , 2 x 2 matrix. 1 3 2
1 8 B  3 1 2 , 3 x 3 matrix.

2 3 1

4. Zero Matrix is a (m x n) matrix which every entry is zero, and denoted by O.

Example 7 0 0 0 0 0 0 0
O  0 0 , O  0 0 , 0 0
0 0 0 0 0 O 
0

USER 2

MATHEMATICS UNIT : FET/FIB
MAT 1064

5. Diagonal Matrix

 a11 a12 a13 a1m 
 
Let A =  a21 a22 a23 a2m 
a32 a33
 a31 a3m 
 
 
am1 am2 am3 amm 

 The diagonal entries of A are a11 , a22 , a33 ,…, amm
 A square matrix which non-diagonal entries are all zero is called a diagonal matrix.

Example 8

(a) A  2 0 1 0 0 a 0 0
0 3 (b) B  0 2 0 (c) C  0 0 0

0 0 3 0 0 b

6. Identity Matrix is a diagonal matrix which all its diagonal entries are 1, denoted by I.

Example 9

(a) A  1 0  I 22 1 0 0
0 1 (b) B  0 1 0  I33

0 0 1

7.

Lower Triangular Matrix is a square matrix and Upper Triangular Matrix is a square matrix and

aij  0 for i  j aij  0 for i  j

 a11 a12 a12  a11 a12 a12 
A  a21  A  a21 
a22 a23  a22 a23 
a31
0 a32 a33  a31 a32 a33 
0
Example 10 3 Example 11

1 0 a 0 0 1 2 3 a b c 
A = 3 2 B = b f 0 A = 0 2 4 B = 0 d 
d e e 
3 2 c 0 0 3
0 0 f 

USER 3

MATHEMATICS UNIT : FET/FIB
MAT 1064

2.2 OPERATIONS ON MATRICES

 ADDITION AND SUBTRACTION OF MATRICES

For m x n matrices, A = [ aij ] and B = [ bij ],

A + B = C = cij mxn , where cij  aij  bij .

A – B = D = dij mxn , where dij  aij  bij .

Note:

The addition or subtraction of two matrices with different orders is not defined. We say the two matrices
are incompatible.

Example 12

Simplify the given quantity for A 1 2 , B  4 3 and C  1 .
3 4  5 6 2

(a) A + B (b) A – B (c) A + C

a) b)

 5 5 Ans :  3 1
Ans :  2 10   2
 8

c) A  C  1 2  1
3 4 2

Since matrix A is of order 2 x 2 and matrix C is of order 2 x 1, the matrices have different orders, thus
A and C are incompatible.

USER 4

MATHEMATICS UNIT : FET/FIB
MAT 1064

 SCALAR MULTIPLICATION
If is a scalar and A  aij  then cA  bij  where bij  caij

Example 13 Example 14

2 4 1 Let A  1 4 and B  3 6
  2 5 3 4 2 . Calculate
Given A  8 5  , find  A .
7 
6 3A  2B.

Solution Solution

  1  2  1  4 3A  2B  3 1 4  2 3 6
 2 2 5 3 4 2
 
1  1 1 
 2 A    2 8  2 5   3 12  6 12
15 8
 1 6 1 7  9  4 
 2 2   
  

= 3 0
 5
   7
 1
 2
 4 
  5 
3
 2

 7 

2

Properties

(a) A  B  B  A ( Commutative )

(b)  A  B  C  A   B  C  ( Associative)

(c) A   A   A  A  O ( O- zero matrix)

(d)     A   A   A  ,  constant

(e)   A  B   A  B

(f)   A    A

USER 5

MATHEMATICS UNIT : FET/FIB
MAT 1064

Exercises 1.1

1. (a) Find matrix A  aij 2x3 if aij  i2 j  j2i

2i  j, i  j
ij, i 
(b) Find matrix B  bij 3x3 if bij  i  2 j, j
i
 j

2. Simplify the given quantity for A  1 2 and B  2 1
3 5 4 9 .

(a) A + B (b) A – B

(c) 2A – 5B (d) 3A + 2B

3. Solve the given equation for the unknown matrix X.

1 2 3 0 0 0
(a) 2X + 6 5 4 = 0 0 0

(b) -2  X  2 6   4 X  0 0
3 3 2 4

Ans :

2 6 12 1 4 5
1. (a) 6 16 30 (b) 0 4 7

1 1 9

3 3  1 1   8 1  7 8
2. (a) 7 14 (b)  1  4 (c) 14  35 (d) 17 33

3. (a)  1 1  232  2 6
 2  (b) 4 5
 5
3 2

USER 6

MATHEMATICS UNIT : FET/FIB
MAT 1064

 Multiplication of Matrices

The product of two matrices A and B is defined only when the number of columns in A is equal to the
number of rows in B.

If the order of A is m  n and the order of B is n  p, then AB has order m  p.

AmnBnp  ABmp
A row and a column must have the same number of entries in order to be multiplied.

 b1 
 
 b2 

R  a1 a2 a3 ... an  and C  ...b3 


 bn 

RC  a1b1  a2b2  a3b3  ...anbn 

Example 15

1 2 3  2 1
Find 2 0 5 3 4
 2 1

1 2 3  2 1  1(2)  2(3)  3(2) 1(1)  2(4)  3(1)  2 12
 2 0 5  3 4  2(2)  0(3)  5(2)  2(1)  0(4)  5(1) = 6
  2 1 = 3 


Example 16

2 5 4  1 2 3 5 
2 0 5  3 
A  and B  5 2 1 5  . Find AB.
4
0 7 

Ans : AB  7 30 11 43 
45 4 10 5 

USER 7

MATHEMATICS UNIT : FET/FIB
MAT 1064

Example 17

Let A  1 2 and B  2  1
 4 3 
 3 2 

Show that AB ≠ BA.

1 2 2 1 (1)2  2(3) (1)(1)  2(2)  4 5
AB =  3 4 3 2  =  3(2)  4(3) = 18 5
3(1)  4(2) 

2 1  1 2 2(1)  (1)3 2(2)  (1)4  5 0 
BA = 3   4 =   =  14
2   3  3(1)  2(3) 3(2)  2(4)   3

Thus, AB ≠ BA.

Properties ( Associativite)

(a) ABC    ABC

(b) A B  C   AB  AC ( Distributive)

Transpose Matrix

The transpose of a matrix A, written as AT, is the matrix obtained by interchanging the rows and columns
of A. That is, the i th column of AT is the i th row of A for all i’s.

If Amn  aij  then AT nm  a ji 

a11 a12 a13  a11 a21 a31 
A  a21   
a22 a23  then AT   a12 a22 a32 

a31 a32 a33 33 a13 a23 a33 33

Properties of Transpose Matrix

 (A ± B)T = AT ± BT

 (AT)T = A

 (AB)T = BTAT

 (kA)T = kAT

USER 8

Example 18 MATHEMATICS UNIT : FET/FIB
2 MAT 1064

Let B  1 then BT  2 1 3 13 USER 9

331

1 3 3 1 2 1
If D  2 5 4 then DT  3 5 3
1 3 533 3 4 533

Example 19

Let A  1 2 B  3 4 and C  1 4
3 4 , 2 1 3 2 .

Show that (a) (A + B )T = AT + BT
(b) (BC)T = CTBT

Solution

1 3 2  4 4 6
(a) A + B = 3  2 4 1 = 5 5

(A + B )T = 4 5
6 5

AT + BT = 1 3  3 2
2 4 4 1

1 3 3 2
= 2  4 4 1

4 5
= 6 5

 (A + B )T = AT + BT

MATHEMATICS UNIT : FET/FIB
MAT 1064

(b) 3 4  1 4T
(BC)T = 2 1 3 2

= 3 12 12  8T
 
 2  3 8  2 

= 15 20T
 
 5 10 

15 5 
= 20 10

CTBT 1 4T  3 4T
= 3 2 2 1

1 3  3 2
= 4 2 4 1

3  12 2  3
= 12  8 8  2

15 5 
= 20 10

 (BC)T = CTBT

USER 10

MATHEMATICS UNIT : FET/FIB
MAT 1064

Exercises 1.2

1 2 4 1 2 3 7 9
2 3 0 , 3 4 , and C  4 1 .Indicate whether the given product
1. Let A  B  2 5 4
6

is defined. If so, give the order of the matrix product. Then compute the product, if possible.

(a) (b) (c)

(d) (e) (f)

 3 1 2 1 3 4
Let A   4  B  1  2 2 2 .
2.  2 0  ,  1 1  and C 

1 

Find ATB (b) BTA (c) (BC)T (d) (A+B)T
(a)
Answers

1. (a) Not defined

19 41 27 
(a) Defined; 2x3 ; 18 29 21 

 5 8 4
(b) Defined; 2x3 ; 11 
18 12 

(c) Not defined

(d) Not defined

(e) Not defined

2. (a) 12 13 (b) 12 1
 1  13 
0  0 

 8  7 5 5  5 3
(c) 10  8 6 (d) 0  2 2

USER 11

MATHEMATICS UNIT : FET/FIB
MAT 1064

2.3 DETERMINANT 0F 2x2 AND 3x3 MATRIX

Determinant of 2 x 2 Matrices

a b
 c 
Given A = d 

Then determinant A = ab = ad – bc

cd

Example 22

2 5 3 2
Given A = 3 8 and B = 5 2 , find A , B , AB , BA .

USER 12

MATHEMATICS UNIT : FET/FIB
MAT 1064

Minor and Cofactor

Let be × matrix,

1. The minor Mij of the element aij is the determinant of the matrix obtained by deleting the ith row
and jth column of .

2. The cofactor Cij of the element aij is Cij = (-1)i+j Mij

 1 2 -1 

A   3 4 2 
 

 1 4 3 

 Minor

M11 is the determinant of the matrix obtained by deleting the first row and first column of A.

1 2 −1
M11 = |3 4 2 | = 4(3)-4(2) = 4

14 3

Similarly

1 2 −1
M32 = |3 4 2 | = 1(2) – (3)(-1) = 5

14 3

Therefore,

 a a a 
 13 
If A =  11 12  , then


 a
 a a 23 
 
21 22

 
 
 a a a 
 31 32 33 

aM11 = 22 a23 aand M32 = 11 a13
a32 a33 a21 a23

Cofactor

Cij = ( - 1 )i +j Mij

Then, C11 = (-1)1+1 M11 = 4 and
C32 = (-1)3+2 M32 = -5

USER 13

Example 23 MATHEMATICS UNIT : FET/FIB
MAT 1064
 2  4  2
A   2 0  (ii) M31 =  4  2 = ( -16 – 0 ) = -16
4  04

 4 3  3 C31 = (-1)3+1(-16) = -16
(iv) M23 = 2  4 = (6 + 16 ) = 22
find
43
i . M12 and C12 C23 = (-1)2+3 (22) = -22
ii. M31 and C31
iii. M22 and C22
iv. M23 and C23

(i) M12 =  2 4 = ( 6 – 16) = -10
4 3

C12 = ( -1 )1+2 (-10) = 10

(iii) M22 = 2  2 = ( -6 + 8 ) = 2
4 3

C22 = (-1)2+2(2) = 2

Determinant of 3 x 3 Matrix

Expansion of the cofactor

A  aij cij ;

i 1, 2, ..., n and j 1, 2, ..., n

By expanding along the first row a11, a12 , a13 By expanding along first column
Elements in 1st row : Elements in 1st column : a11, a21, a31

A  a11c11  a12c12  a13c13 A  a11c11  a21c21  a31c31
A  a11m11  a12m12  a13m13
A  a11m11  a21m21  a31m31

USER 14

MATHEMATICS UNIT : FET/FIB
MAT 1064

Example 25

3 1 4 
Let A  1 2 
7  , find A by e xpanding along ;

5 1 10

a) sec ond row

b) first column

a) b)

3 1 4  3 1 4 
1  2 7  1  2 7 
5 1 10 5 1 10

1 4  (2) 3 43 1  2 1 4
| A | 1  (7) 1 | A | (3)  (1)  (5)
1 10 5 10 1 10
10 5



  3(2)(10)  (7)(1) (1)(1)(10)  (1)(4)
 (5)(1)(7)  (4)(2)
 28


 28

USER 15

MATHEMATICS UNIT : FET/FIB
MAT 1064

USER 16

MATHEMATICS UNIT : FET/FIB
MAT 1064

USER 17

MATHEMATICS UNIT : FET/FIB
MAT 1064

USER 18

MATHEMATICS UNIT : FET/FIB
MAT 1064

Example 28

 1 1 1 
Given that A   a b c  and A  3. By using the properties of determinant, evaluate

 a2 b2 c2 

3  2a2 3  2b2 3  2c2 a 1 a2
b) b 1 b2
a) a b c
a2 b2 c2 c 1 c2

2a 2b 2c d) A3
c) a2 b2 c2

222

3  2a2 3  2b2 3  2c2 3 3 3  2a2  2b2  2c2

a) a b c  a b c  a b c

a2 b2 c2 a2 b2 c2 a2 b2 c2

111 a2 b2 c2

 3 a b c   2 a b c

a2 b2 c2 a2 b2 c2

 3 A   20  33  0

9

USER 19

MATHEMATICS UNIT : FET/FIB
MAT 1064

a 1 a2 1 a a2 111

b) b 1 b2   11 b b2   1 a b c  (1) A  3

c 1 c2 1 c c2 a2 b2 c2

Ans : c) 12 d) 27

EXERCISES 1.3 :

 2 5 1 
1) Find the determinant of A  3 0 
1  by using expansion of the cofactor.

 2 5 4

2) Find the determinant for these matrices by using the method above:

(a) A= 6  3
2 3 

(b) B =  2 3 1
 
 0 2 4 

 2 5 6 

t  1 5 3 
3) The matrix N is given by N    3 1   1 . If N  181, M31  7
2t  3
  2  

and C23  11 , determine the positive values of t ,  and  .

ANSWERS:

1) 25

2) a. 24 b. -44

3)   3, t  2,   5

USER 20

MATHEMATICS UNIT : FET/FIB
MAT 1064

Adjoint Matrix

 Let C  cij be the cofactor matrix of A.

Adjoint of matrix A (adj A) is defined as the transpose of the cofactor matrix that is

   adj A  CT  cij T  c ji

Remember: Cofactor , cij = (-1)i+j mij,   m11 m12 m13 
C  m21 m22
m32 m23 
m31 
m33 

Example 29

1 2 3
Given A  3 2 4 . Find the adjoint of A.

1 1 3

24 34 32
c11  1 2 c12   1  5 c13  1 1
3 3 1

c21   c22   c23  

c31   c32   c33  

2 5 1 T
adj A  3
0 1  NOTE : adj A = CT
 2 
5 4

 2 3 2 

adj A  5 0 5 

 1 1 4

Example 30

1 2 2 15 0 10
Given P  2 10 5 . Find the adjoint of P.
adj P   1 1 1 
1 3 3  

4 1 6 

USER 21

2.4 INVERSE OF 2x2 and 3x3 MATRIX MATHEMATICS UNIT : FET/FIB
MAT 1064
SINGULARITY OF MATRICES
If A  0 ,
If A  0 ,
~ A is a singular matrix
~ A is a non-singular matrix ~ Inverse matrix does not exist
~ Inverse matrix exists

There are 2 methods to obtain inverse of matrices:

(a) Adjoint Method; A-1 = 1 adj A *( Remember! A1  1 )
AA

(b) Using the Property of AB  kI

Finding Inverse By Using Adjoint Method

The inverse of a matrix A is denoted by A1  1 adj A , given that A  0 .
A

Inverse of 2 x 2 Matrix

Let A a b , then A1 is given by
c d 

A1  1 d b Note A1  1
 c  A
ad bc a 

Example 31

Find the inverse matrix for A 3 1
5 4

4  1 
 7 
A 1  7 

 5 3
7 7 

USER 22

MATHEMATICS UNIT : FET/FIB
MAT 1064

Inverse of 3 x 3 Matrix

Example 32

 1 3 2
 2 2
Find the inverse matrix of B   0

2 1 0

Determinant B = |B| =

 22 02 0 2
 1 0  
 2 1 
32 2 0
  12 1 3  2 4 4
 2 0   = 2 4 5
Cofactor B   1 0 1 2 1  2 2 
02 2  2
 32 
 22 13 
 
 02 

 
 
Adjoint B =  




 2 2 2  1 1 1
1 4 2  
B1   2  4 4   2 2 1
5 5
2   
2 1
 2

If AB  I where A and B are square matrices, then B is called the inverse matrix of A and is written as
A1 . Thus AA1  A1A  I

Example 33

1 2 3  1 1 1
Given A  2 3 4 and B  10 4 
2  . It is known that AB  kI , where k is a constant and

1 5 7  7 3 1

I is an 33 matrix. Find k and hence deduce A1 .

1  1 1 1
2 10 
A 1   7 4 2 
3
1

USER 23

MATHEMATICS UNIT : FET/FIB
MAT 1064

2.5 SYSTEM OF LINEAR EQUATIONS WITH 3 VARIABLES

Using the Inverse Matrix to solve AX = B

If A is a n x n square matrix that has an inverse A-1, X is a variable matrix and B is a known matrix, both
with n rows, then the solution of matrix equations AX = B is given by

X = A-1 B

Proof : A X = B *( 3 x 3 square matrix)

A-1 ( A X ) = A-1 B

( A-1A ) X = A-1B

I X = A-1 B

X = A-1 B

Example 34 A1 
Solve the following system of equations by using
the inverse matrix

3x1  x2  2x3  11

3x1  2x2  2x3  10

x1  x3 5 AX=B
A-1 ( A X ) = A-1 B
Convert this to a matrix equation of the form ( A-1A ) X = A-1B
AX = B
I X = A-1 B
X = A-1 B

    
    
     

      x1   2 1  2 11
  = 1  10
 x2  1 0 

det A  x3   2 1 3   5 

  T  22 10 10 
   
   =  11  10  0 
adjA   
     22  10  15
  
   2
  =  1

 3 

 Therefore x1  2, x2  1 and x3  3

adjA  



USER 24

MATHEMATICS UNIT : FET/FIB
MAT 1064

Example 35: Example 36:
Solve the following system of equations using
1 3 9  1 3 0 the inverse matrix
Given A = 5 1   1 3
3  B=  2 9x1  4x2  x3  17
x1  2x2  6x3  14
1 3 7  1 0 1 x1  6x2  4

Find AB and A-1. Hence, solve the following linear

equations.

x5y  z  7

3x  y  3z  5

9x  3y  7z  1

= 1, = 1 and = 1 x1= -2, x2 = 1 and x3 = -3

USER 25

MATHEMATICS UNIT : FET/FIB
MAT 1064

Cramer’s Rule

Step 1 : Find the determinant of Matrix A, A

Step 2 : Replacing the column of A with n x 1 matrix B

a11 a12 a13   x1  b1 
a21    b2 
a22 a23   x2  = 

a31 a32 a33  x3  b3 

A X =B

Step 3 : Then the solution is given by

b1 a12 a13 a11 b1 a13 a11 a12 b1
b2 a22 a23 a21 b2 a23 a21 a22 b2
x1 = b3 a23 a33 x2 = b31 b3 a33 x3 = b31 a23 b3

A A A

Example

Solve the following system of linear equations using Cramer’s Rule.

x1  x2  2x3  3
x1  x2  3x3  11
2x1  3x2  x3  9

Solution

First, the system can be expressed into the matrix form as AX = B

    
    
     

    

Then , find the determinant of A , A

A=

=
=
= 19

USER 26

3 -1 2 MATHEMATICS UNIT : FET/FIB
- 11 1 - 3 MAT 1064

9 31 USER 27

x1 =

19
= 9(3  2)  3(9  22) 1(3 11)

19
= 9  39  8

19
=  38

19
= -2

132
1 - 11 - 3
29 1

x2 =

19

=
=
=3

1 -1 3
1 1 - 11
239

x3 =

19

=
=
=4

So, x1  2 , x2  3 and x3  4

MATHEMATICS UNIT : FET/FIB
MAT 1064

Exercise 1.4

1) A shop sales three types of toys A, B and C. Total price from the sales of 7 units of toys A, 3 unit
of toys B and 4 unit of toys C is RM670. The total price from the sales of 5 unit of toys A, 2 unit
toys B and 3 unit toys C is RM480. The total price from the sales of a unit toy A, 2 unit of toys B
and 3 unit of toys C is RM320.
a. If x, y and z represent the price in RM for every toys A, B and C respectively, determine
a system of linear equation based on the above information.
b. Write down the system of linear equation in matrix form.
c. By using Cramer’s Rule, calculate the price in RM for every toys of A, B and C.
d. If the shop gives 1% discount for every unit of toys C, what is the total price in RM from
the sales of toys C.

2) A machine produces 3 types of biscuits A, B and C . The ingredients for each type of biscuit are

shown in the table below.

Ingredients Types of Biscuit Stock
ABC (grams)
Flour (grams) 734
Sugar (grams) 523 6700
Milk (grams) 123 4800
3200

There is now left a stock 6.7kg of flours, 4.8kg of sugars and 3.2kg of milk. Let x, y and z be
the number of biscuits of type A, B and C produced respectively.

a. Obtain a system of linear equation to represent the above information.
b. Express the system of linear equations in (a) as a matrix equation.

c. Find the number of biscuits A, B and C .

d. If for each type of biscuits, the amount of flour, sugar and milk is increased by 3 grams, 2 grams
and a gram respectively, form a new matrix equation.

USER 28

MATHEMATICS UNIT : FET/FIB
MAT 1064

Ans : 2.
1. a)

7x  3y  4z  670 7 3 4x 6700
a) 5x  2 y  3z  480 b) 5 2 3 y  4800

x  2 y  3z  320 1 2 3 z  3200

7 3 4x 670 A  400units, B  500units
b) 5 2 3 y  480
c)
1 2 3 z  320
C  600units
c) A=RM40 , B=RM50,
C=RM60 10 6 7x 6700
 5  y 4800
d) RM594 d)  7 4  

 2 3 4 z 1200 

USER 29