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CSNB143 â€“ Discrete Structure

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CSNB143 â€“ Discrete Structure

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.

Introduction

- An array of numbers arranged in m horizontal rows and n vertical columns:
- Ex 1:
A = a11a12â€¦â€¦.a1n

a21a22â€¦â€¦.a2n

......

......

......

am1am2â€¦â€¦.amn

- The ith row of A is [ai1, ai2, ai3, â€¦ain]; 1 ï‚£iï‚£m
- The jth column of A is a1j
a2j ; 1 ï‚£jï‚£n

a3j

.

.amj

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 andjth 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 iï‚¹ j, is called a diagonal matrix.
- Ex 2:
A = 8 0 0 0

0 3 0 0

0 0 7 0

0 0 0 1

- Two m x n matrices A and B, A = [aij] and B = [bij], are said to be equal if aij = bij for 1 ï‚£iï‚£m, 1 ï‚£jï‚£n; that is, if corresponding elements are the same.
- Ex 3:
A = a 5 3B = 1 5 x

2 7 -1y 7 -1

3 b 03 4 0

- So, if A = B, then a = 1, x = 3, y = 2, b = 4.

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 ï‚£iï‚£m, 1 ï‚£j ï‚£n.

- C is obtained by adding the corresponding elements of A and B.
- Ex 4:
A = 1 5 3B = 2 0 3C = 3 5 6

2 7 -1 +6 1 3 =8 8 23 4 0-3 1 90 5 9

- 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).

Exercise 1:

a) Identify which matrices that the summation process can be done.

b) Compute C + G, A + D, E + H, A + F.

A = 2 1 B = 2 1 3C = 7 2D = 3 3

4 84 5 74 22 5

1 5

E = 2 -3 7F = -2 -1G = 4 3H = 1 2 3

0 4 7 -4 -85 14 5 6

3 1 2-1 07 8 9

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 + â€¦ + aipbpj; 1 ï‚£iï‚£n, 1 ï‚£j ï‚£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.

Ex 5:

A = 2 3 -4B = 3 1

1 2 3-2 2

2 x 3 5 -3

3 x 2

AB = 2(3) + 3(-2) + -4(5)2(1) + 3(2) + -4(-3)

1(3) + 2(-2) + 3(5)1(1) + 2(2) + 3(-3)

= 6 â€“ 6 â€“ 202 + 6 + 12

3 â€“ 4 + 151 + 4 â€“ 9

= -2020

14-4 2 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.

- 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:
- nï‚¹m, then BA cannot be produced.
- n = m, pï‚¹m @ n, then we can get BA but the size will be different from AB.
- n = m= p, A ï‚¹ B, then we can get BA, the size of BA and AB is the same, but AB ï‚¹ BA.
- n = m = p, A = B, then we can get BA, the size of BA and AB is the same, and AB = BA.

A B AB B ABA

(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 33 X 3

2 X 2 2 X 2 2 X 2 2 X 2 2 X 22 X 2

2 1 3 19 5 3 1 2 1 8 6

2 3 3 3 15 11 3 3 2 312 12

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:
1 01 0 01 0 0 0

0 10 1 00 1 0 0

0 0 00 0 1 0

0 0 0 1

- Theorems involved are:
- A(BC) = (AB)C.
- A(B + C) = AB + AC.
- (A + B)C = AC + BC.
- IA = AI = A.

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 ï‚£iï‚£m, 1 ï‚£jï‚£n

- It is called transposition matrix for A.
- Ex 8:
A = 2 -3 5AT = 2 6

6 1 3-3 1

5 3

- Theorems involved are:
- (AT)T = A
- (A + B)T = AT + BT
- (AB)T = BTAT

- 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 = 12 -3B = 1 2 -3

2452 4 0

-3563 2 1

Symmetric Not Symmetric, why?

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 ïƒš B, will produce a matrix C = [cij], where

cij = 1if aij = 1 OR bij = 1

0if aij = 0 AND bij = 0

- Meet
Meetfor A and B, both with the same dimension, written as A ïƒ™ B, will produce matrix D = [dij] where

dij= 1if aij = 1 AND bij = 1

0 if aij = 0 OR bij = 0

- Ex 10: A = 1 0 1B = 1 1 0
- 0 1 10 0 1
- 1 1 00 1 0
- 0 1 01 1 0
- A ïƒš B = 1 1 1A ïƒ™ B = 1 0 0
- 0 1 10 0 1
- 1 1 00 1 0
- 1 1 00 1 0

- 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 âŠ™ B, producing C, where:
- cij = 1 if aik = 1 AND bkj = 1; 1 ï‚£kï‚£p.
0other than that

- It is using the same way as normal matrix product.

- Ex 11:
- A = 1 0 0 0B = 1 1 0
- 0 1 1 00 1 0
- 1 0 1 11 1 0
- 3 x 40 0 1
4 x 3

A âŠ™ B = 1 + 0 + 0 + 01 + 0 + 0 + 00 + 0 + 0 + 0

0 + 0 + 1 + 00 + 1 + 1 + 00 + 0 + 0 + 0

1 + 0 + 1 + 01 + 0 + 1 + 00 + 0 + 0 + 1

- A âŠ™ B = 1 1 0
1 1 0

1 1 1

3 x 3

- Exercise 3:
- A = 1 0 0 0B = 0 1 0 0C = 0 0 1 0
0 1 1 00 0 1 11 0 0 0

0 0 0 10 1 0 11 1 0 0

1 1 0 00 0 1 01 1 1 0

- Find:
- A ïƒš B
- A ïƒ™ B
- A âŠ™ B
- A ïƒš C
- A ïƒ™C
- A âŠ™ C
- B ïƒš C
- B ïƒ™ C
- B âŠ™ C