CSNB 143 Discrete Mathematical Structures

1. CSNB 143 Discrete Mathematical Structures Chapter 4 – Matrix

2. 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.

3. 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

4. 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].

5. 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.

6. 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.

7. 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.

8. 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).

9. 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

10. 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

11. 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.

12. 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.

13. 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)

14. = 6 – 6 – 20 2 + 6 + 12 3 – 4 + 15 1 + 4 – 9 = -20 20 14 -4 2 x 2

15. Exercise 2: • Identify which matrices that the product process can be done. List all pairs. • Compute CA, AD, EG, BE, HE.

16. 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.

17. 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

18. 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.

19. 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

20. 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 -3 B = 1 2 -3 245 2 4 0 -356 3 2 1 Symmetric Not Symmetric, why?

21. 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

22. 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 1 1 0 0 1 0

23. 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.

24. 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

25. MATRIX • 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