Chapter 4 – Matrix

CSNB 143 Discrete Mathematical

Structures

MatrixStudents 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.

MatrixAn array of numbers arranged in m horizontal rows

and n vertical columns: A = a11 a12 a13 ……. a1n

a21 a22 a23 …….. a2n

… … … …………

am1 am2 am3 …… 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

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].

Ex 2:A = 8 0 0 0

0 3 0 00 0 7 00 0 0 1

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.

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 5 3 B = 1 5

x2 7 -1 y 7 -13 b 0 3 4 0

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

Matrix summationIf 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.

A = 1 5 3 B = 2 0 3 2 7 -1 6 1 3 3 4 0 -3 1 9

C = 3 5 6 8 8 4 0 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 3 C = 7 2

4 8 4 5 7 4 2

1 5

D = 3 3 E = 2 -3 7 F = -2 -1 2 5 0 4 7 -4 -8

3 1 2

G = 4 3 H = 1 2 3 5 1 4 5 6 -1 0 7 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 ProductIf 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.

Ex 5:A = 2 3 -4 B = 3 1

1 2 3 -2 22 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 – 6 – 20 2 + 6 + 123 – 4 + 15 1 + 4 – 9

= -20 2014 -4

2 x 2

Exercise 2:Identify which matrices that the product

process can be done. List all pairs.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:

n m, then BA cannot be produced. n = m, p m @ 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.

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 22 X 2

2 1 3 1 9 5 3 1 2 1 8 6

2 3 3 3 15 11 3 3 2 3 12 12

MATRIXIdentity 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 0 1 0 0 1 0 0 0

0 1 0 1 0 0 1 0 0

0 0 1 0 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 aij

T = aji; 1 i m, 1 j n  It is called transposition matrix for A.Ex 8:

A = 2 -3 5 AT = 2 66 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 = 1 2 -3 B = 1 2 -3

2 4 5 2 4 0

-3 5 6 3 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:

a) Join byGiven 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 = 10 if aij = 0 AND bij = 0

b)MeetMeet 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 = 10 if aij = 0 OR bij = 0

MATRIXEx 10: A = 1 0 1 B = 1 1 0

0 1 1 0 0 11 1 0 0 1 00 1 0 1 1 0

 

A B = 1 1 1 A B = 1 0 0 0 1 1 0 0 1 1 1 0 0 1 0 1 1 0 0 1 0

MATRIXc) 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.

MATRIXEx 11:

A = 1 0 0 0 B = 1 1 0

0 1 1 0 0 1 0

1 0 1 1 1 1 0

3 x 4 0 0 1

4 x 3

A ⊙ B = 1 + 0 + 0 + 0 1 + 0 + 0 + 0 0 + 0 + 0 + 0

0 + 0 + 1 + 0 0 + 1 + 1 + 0 0 + 0 + 0 + 0

1 + 0 + 1 + 0 1 + 0 + 1 + 0 0 + 0 + 0 + 1

  A ⊙ B = 1 1 0

1 1 0

1 1 1

3 x 3

MATRIX Exercise 3: A = 1 0 0 0 B = 0 1 0 0 C = 0 0 1

00 1 1 0 0 0 1 1 1 0 0

00 0 0 1 0 1 0 1 1 1 0

01 1 0 0 0 0 1 0 1 1

1 0

Find:a) A Bb) A Bc) A ⊙ Bd) A Ce) A Cf) A ⊙ C g) B Ch) B Ci) B ⊙ C