CSNB 143 Discrete Mathematical Structures Chapter 4 – Matrix

Matrix 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.

Matrix An 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 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 0 0 0 7 0 0 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 x 2 7 -1 y 7 -1 3 b 0 3 4 0 So, if A = B, then a = 1, x = 3, y = 2, b = 4.

Matrix 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.

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 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.

Ex 5: A = 2 3 -4 B = 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 – 6 – 20 2 + 6 + 12 3 – 4 + 15 1 + 4 – 9 = -20 20 14 -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) 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 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

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: 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 aijT = aji; 1  i  m, 1  j  n   It is called transposition matrix for A. Ex 8: A = 2 -3 5 AT = 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 = 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: 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

MATRIX Ex 10: A = 1 0 1 B = 1 1 0 0 1 1 0 0 1 1 1 0 0 1 0 0 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

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.

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