1. CSNB143 – Discrete Structure
MATRIX

2. MATRIX
Learning Outcomes
Students should be able to read matrix and its entries without difficulties.
Students should understand all matrices operations.
Students should be able to differentiate different matrices and operations by different matrix.
Students should be able to identify Boolean matrices and how to operate them.

3. MATRIX
Introduction
An array of numbers arranged in m horizontal rows and n vertical columns:
Ex 1:
A = a11 a12 ……. a1n
a21 a22 ……. a2n
am1 am2 ……. amn
The ith row of A is [ai1, ai2, ai3, …ain]; 1  i  m
The jth column of A is a1j
a2j ; 1  j  n
a3j .
.amj

4. MATRIX Diagonal matrix
We say that A is a matrix m x n. If m = n, then A is a square matrix of order n, and a11, a22, a33, ..ann form the main diagonal of A.
aij which is in the ith row and jth column, is said to be the i, jth element of A or the (i, j) entry of A, often written as A = [aij].
A square matrix A = [aij], for which every entry off the main diagonal is zero, that is aij = 0 for i  j, is called a diagonal matrix.
Ex 2:
A =

5. MATRIX
Two m x n matrices A and B, A = [aij] and B = [bij], are said to be equal if aij = bij for 1  i  m, 1  j  n; that is, if corresponding elements are the same.
Ex 3:
A = a B = x y
3 b
So, if A = B, then a = 1, x = 3, y = 2, b = 4.

6. MATRIX Matrices Summation
If A = [aij] and B = [bij] are m x n matrices, then the sum of A and B is matrix C = [cij], defined by
cij = aij + bij; 1  i  m, 1  j  n.
C is obtained by adding the corresponding elements of A and B.
Ex 4:
A = B = C = =
The sum of the matrices A and B is defined only when A and B have the same number of rows and the same number of columns (same dimension).

7. MATRIX
Exercise 1:
a) Identify which matrices that the summation process can be done.
b) Compute C + G, A + D, E + H, A + F.
A = B = C = D =
1 5
E = F = G = H =

8. MATRIX
A matrix in when all of its entries are zero is called zero matrix, denoted by 0.
Theorems involved in summation :
A + B = B + A.
(A + B) + C = A + (B + C).
A + 0 = 0 + A = A.
Matrices Product
If A = [aij] is an m x p matrix and B = [bij] is a p x n matrix, then the product of A and B, denoted AB, will produce the m x n matrix C = [cij], defined by
cij = ai1b1j + ai2b2j + … + aipbpj; 1  i  n, 1  j  m
That is, elements ai1, ai2, .. aip from ith row of A and elements b1j, b2j, .. bpj from jth column of B, are multiplied for each corresponding entries and add all the products.

9. MATRIX
Ex 5:
A = B =
2 x
3 x 2
AB = 2(3) + 3(-2) (5) 2(1) + 3(2) + -4(-3)
1(3) + 2(-2) (5) 1(1) + 2(2) + 3(-3)
= 6 – 6 –
3 – – 9 =
x 2
Exercise 2:
a) Identify which matrices that the product process can be done. List all pairs.
b) Compute CA, AD, EG, BE, HE.

10. MATRIX
If A is an m x p matrix and B is a p x n matrix, in which AB will produce m x n, BA might be produce or not depends on:
n  m, then BA cannot be produced.
n = m, p  n, then we can get BA but the size will be different from AB.
n = m= p, A  B, then we can get BA, the size of BA and AB is the same, but AB  BA.
n = m = p, A = B, then we can get BA, the size of BA and AB is the same, and AB = BA.

11. A B AB B A BA (m x p) (p x n) (m x n) (p x n) (m x p)
A B AB B A BA (m x p) (p x n) (m x n) (p x n) (m x p) ? 2 x 3 3 x 4 2 x 4 3 x 4 2 x 3 X 2 x 3 3 x 2 2 x 2 3 X 2 2 X 3 3 X 3 2 X 2 2 X 2 2 X 2 2 X 2 2 X 2 2 X

12. MATRIX Identity matrix
Let say A is a diagonal matrix n x n. If all entries on its diagonal are 1, it is called identity matrix, ordered n, written as I.
Ex 7:
Theorems involved are:
A(BC) = (AB)C.
A(B + C) = AB + AC.
(A + B)C = AC + BC.
IA = AI = A.

13. MATRIX Transposition Matrix
If A = [aij] is an m x n matrix, then AT = [aij]T is a n x m matrix, where
aijT = aji; 1  i  m, 1  j  n
It is called transposition matrix for A.
Ex 8:
A = AT =
5 3
Theorems involved are:
(AT)T = A
(A + B)T = AT + BT
(AB)T = BTAT

14. MATRIX
Matrix A = [aij] is said to be symmetric if AT = A, that is aij = aji,
A is said to be symmetric if all entries are symmetrical to its main diagonal.
Ex 9:
A = B =
Symmetric Not Symmetric, why?

15. MATRIX Boolean Matrix and Its Operations
Boolean matrix is an m x n matrix where all of its entries are either 1 or 0 only.
There are three operations on Boolean:
Join by
Given A = [aij] and B = [bij] are Boolean matrices with the same dimension, join by A and B, written as A  B, will produce a matrix C = [cij], where
cij = 1 if aij = 1 OR bij = 1
0 if aij = 0 AND bij = 0
Meet
Meet for A and B, both with the same dimension, written as A  B, will produce matrix D = [dij] where
dij = 1 if aij = 1 AND bij = 1
0 if aij = 0 OR bij = 0

16. MATRIX
Ex 10: A = B =
A  B = A  B =

17. MATRIX Boolean product
If A = [aij] is an m x p Boolean matrix, and B = [bij] is a p x n Boolean matrix, we can get a Boolean product for A and B written as A ⊙ B, producing C, where:
cij = 1 if aik = 1 AND bkj = 1; 1  k  p.
0 other than that
It is using the same way as normal matrix product.

18. MATRIX
Ex 11:
A = B =
3 x
4 x 3
A ⊙ B =
A ⊙ B =
3 x 3

19. MATRIX Exercise 3: A = 1 0 0 0 B = 0 1 0 0 C = 0 0 1 0
Find:
A  B
A  B
A ⊙ B
A  C
A  C
A ⊙ C
B  C
B  C
B ⊙ C