- 1. Matrices Lecture 17, CMSC 56 Allyn Joy D. Calcaben
- 2. Matrices Definition 1 A matrix is a rectangular array of numbers. A matrix with m rows and n columns is called an m × n matrix. The plural of matrix is matrices. A matrix with the same number of rows as columns is called square. Two matrices are equal if they have the same number of rows and the same number of columns and the corresponding entries in every position are equal.
- 3. Example The matrix is a 3 x 2 matrix 1 1 0 2 1 3
- 4. Matrices Definition 2 Let m and n be positive integers and let a11 a12 . . . a1n a21 a22 . . . a2n A = . . . . . . . . . . . . am1 am2 . . . amn
- 5. Matrices Definition 2 The ith row of A is the 1 x n matrix [ ai1, ai2, . . . , ain] The jth column of A is the m x 1 matrix a1j a2j . . amj
- 6. Matrices Definition 2 The (i, j)th element or entry of A is the element aij, that is, the number in the ith row and jth column of A. A convenient shorthand notation for expressing the matrix A is to write A = [aij], which indicates that A is the matrix with its (i, j)th element equal to aij a11 a12 . . . a1n a21 a22 . . . a2n A = . . . . . . . . . . . . am1 am2 . . . amn
- 8. Matrix Addition Definition 3 Let A = [ aij ] and B = [ bij ] be m × n matrices. The sum of A and B, denoted by A + B, is the m × n matrix that has aij + bij as its (i, j )th element. In other words, A + B = [aij + bij ].
- 9. Example What is the sum of 2 matrices shown below? 1 0 -1 A = 2 2 -3 3 4 0 3 4 -1 B = 1 -3 0 -1 1 2
- 10. Solution What is the sum of 2 matrices shown below? 1 0 -1 A + B = 2 2 -3 3 4 0 3 4 -1 + 1 -3 0 -1 1 2 4 4 -2 = 3 -1 -3 2 5 2
- 11. Matrix Multiplication Definition 4 Let A be an m × k matrix and B be a k × n matrix. The product of A and B, denoted by AB, is the m × n matrix with its (i, j)th entry equal to the sum of the products of the corresponding elements from the ith row of A and the jth column of B. In other words, if AB = [ cij ], then cij = ai1 b1j + ai2 b2j + · · · + aik bkj .
- 12. Example Let and Find AB if it is defined. 1 0 4 A = 2 1 1 3 1 0 0 2 2 2 4 B = 1 1 3 0
- 13. Solution Find AB if it is defined. 1 0 4 AB = 2 1 1 3 1 0 0 2 2 2 4 * 1 1 3 0 14 4 = 8 9 7 13 8 2
- 14. Matrix Multiplication Matrix multiplication is not commutative. That is, if A and B are two matrices, it is not necessarily true that AB and BA are the same. In fact, it may be that only one of these two products is defined. For instance, if A is 2 × 3 and B is 3 × 4, then AB is defined and is 2 × 4; however, BA is not defined, because it is impossible to multiply a 3 × 4 matrix and a 2 × 3 matrix.
- 15. Matrix Multiplication In general, suppose that A is an m × n matrix and B is an r × s matrix. Then AB is defined only when n = r and BA is defined only when s = m. Moreover, even when AB and BA are both defined, they will not be the same size unless m = n = r = s. Hence, if both AB and BA are defined and are the same size, then both A and B must be square and of the same size. Furthermore, even with A and B both n × n matrices, AB and BA are not necessarily equal.
- 16. ¼ Sheet of Paper Let and Does AB = BA ? 1 1 A = 2 1 2 1 B = 1 1
- 17. Solution Does AB = BA ? 3 2 = 5 3 1 1 AB = 2 1 2 1 * 1 1
- 18. Solution Does AB = BA ? 3 2 = 5 3 1 1 AB = 2 1 2 1 * 1 1 4 3 = 3 2 2 1 BA = 1 1 1 1 * 2 1
- 19. Solution Does AB = BA ? Therefore, AB ≠ BA 3 2 = 5 3 1 1 AB = 2 1 2 1 * 1 1 4 3 = 3 2 2 1 BA = 1 1 1 1 * 2 1
- 20. Transposes and Powers of Matrices
- 21. Identity Matrix Definition 5 The identity matrix of order n is the n x n matrix In = [ ẟij ], where ẟij = 1 if i = j and ẟij = 0 if i ≠ j. Hence, 1 0 . . . 0 0 1 . . . 0 In = . . . . . . . . . . . . 0 0 . . . 1
- 22. Transpose Matrix Definition 6 Let A = [ aij ] be an m x n matrix. The transpose of A, denoted by At, is the n x m matrix obtained by interchanging the rows and columns of A. In other words, if At = [ bij ], then bij = aji for i = 1, 2, . . ., n and j = 1, 2, . . ., m.
- 23. Example What is the transpose of the matrix shown below? 1 0 -1 A = 2 2 -3 3 4 0
- 24. Solution What is the transpose of the matrix shown below? 1 0 -1 A = 2 2 -3 3 4 0 1 2 3 At = 0 2 4 -1 -3 0
- 25. ¼ Sheet of Paper What is the transpose of the matrix shown below? 2 4 A = 1 1 3 0
- 26. Solution What is the transpose of the matrix shown below? 2 4 A = 1 1 3 0 At = 2 1 3 4 1 0
- 27. Remember A square matrix A is called symmetric if A = At. Thus, A = [ aij ] is symmetric if aij = aji for all i and j with 1 ≤ i ≤ n and 1 ≤ j ≤ n.
- 28. Zero – One Matrix
- 29. Zero – One Matrix A matrix all of whose entries are either 0 or 1 is called zero-one matrix. Algorithms using these structures are based on Boolean arithmetic with zero-one matrices. This arithmetic is based on the Boolean operations V and ꓥ, which operate on pairs of bits, defined by 1 if b1 = b2 = 1 b1 ꓥ b2 = 0 otherwise, 1 if b1 = 1 or b2 = 1 b1 V b2 = 0 otherwise,
- 30. Join Definition 8.1 Let A = [ aij ] and B = [ bij ] be m × n zero – one matrices. The join of A and B is the zero–one matrix with (i, j)th entry aij ∨ bij, and denoted by A ∨ B.
- 31. Example Find the join of the zero-one matrices. A = 1 0 1 0 1 0 B = 0 1 0 1 1 0
- 32. Solution Find the join of the zero-one matrices. The join of A and B is A = 1 0 1 0 1 0 B = 0 1 0 1 1 0 A = 1 V 0 0 V 1 1 V 0 0 V 1 1 V 1 0 V 0 = 1 1 1 1 1 0
- 33. Meet Definition 8.2 Let A = [ aij ] and B = [ bij ] be m × n zero – one matrices. The meet of A and B is the zero–one matrix with (i, j)th entry aij ꓥ bij, and denoted by A ꓥ B.
- 34. Example Find the meet of the zero-one matrices. A = 1 0 1 0 1 0 B = 0 1 0 1 1 0
- 35. Solution Find the meet of the zero-one matrices. The meet of A and B is A = 1 0 1 0 1 0 B = 0 1 0 1 1 0 A = 1 ꓥ 0 0 ꓥ 1 1 ꓥ 0 0 ꓥ 1 1 ꓥ 1 0 ꓥ 0 = 0 0 0 0 1 0
- 36. Boolean Product Definition 9 Let A = [ aij ] be a m x k zero – one matrix and B = [ bij ] be a k × n zero – one matrices. The Boolean product of A and B, and denoted by A ⨀ B, is the m x n matrix with (i, j)th entry cij where cij = (ai1 ꓥ b1j) V (ai2 ꓥ b2j ) V · · · V (aik ꓥbkj).
- 37. Example Find the Boolean product of A and B, where 1 0 A = 0 1 1 0 B = 1 1 0 0 1 1
- 38. Solution Find the Boolean product of A and B, where The Boolean Product A ⨀ B is given by 1 0 A = 0 1 1 0 B = 1 1 0 0 1 1 A ⨀ B = ( 1 ꓥ 1 ) V ( 0 ꓥ 0 ) ( 1 ꓥ 1 ) V ( 0 ꓥ 1) ( 1 ꓥ 0 ) V ( 0 ꓥ 1) ( 0 ꓥ 1 ) V ( 1 ꓥ 0 ) ( 0 ꓥ 1 ) V (1 ꓥ 1) ( 0 ꓥ 0 ) V (1 ꓥ 1) ( 1 ꓥ 1 ) V ( 0 ꓥ 0 ) ( 1 ꓥ 1 ) V ( 0 ꓥ 1) ( 1 ꓥ 0 ) V ( 0 ꓥ 1)
- 39. Solution Find the Boolean product of A and B, where The Boolean Product A ⨀ B is given by 1 0 A = 0 1 1 0 B = 1 1 0 0 1 1 A ⨀ B = 1 V 0 1 V 0 0 V 0 0 V 0 0 V 1 0 V 1 1 V 0 1 V 0 0 V 0
- 40. Solution Find the Boolean product of A and B, where The Boolean Product A ⨀ B is given by 1 0 A = 0 1 1 0 B = 1 1 0 0 1 1 A ⨀ B = 1 V 0 1 V 0 0 V 0 0 V 0 0 V 1 0 V 1 1 V 0 1 V 0 0 V 0 = 1 1 0 0 1 1 1 1 0
- 41. Boolean Power Definition 10 Let A be a square zero–one matrix and let r be a positive integer. The rth Boolean power of A is the Boolean product of r factors of A. The rth Boolean product of A is denoted by A[r]. Hence A[r] = A ⨀ A ⨀ A ⨀ . . . ⨀ A
- 42. Boolean Power Definition 10 Let A be a square zero–one matrix and let r be a positive integer. The rth Boolean power of A is the Boolean product of r factors of A. The rth Boolean product of A is denoted by A[r]. Hence A[r] = A ⨀ A ⨀ A ⨀ . . . ⨀ A r times
- 43. Example Find A[n] for all positive integers n. 0 0 1 A = 1 0 0 1 1 0
- 44. Solution Find A[n] for all positive integers n. 0 0 1 A = 1 0 0 1 1 0 1 1 0 A[2] = A ⨀ A 0 0 1 1 0 1
- 45. Solution Find A[n] for all positive integers n. 0 0 1 A = 1 0 0 1 1 0 1 0 1 A[3] = A[2] ⨀ A 1 1 0 1 1 1 1 1 0 A[2] = A ⨀ A 0 0 1 1 0 1
- 46. Solution Find A[n] for all positive integers n. 0 0 1 A = 1 0 0 1 1 0 1 0 1 A[3] = A[2] ⨀ A 1 1 0 1 1 1 1 1 1 A[4] = A[3] ⨀ A 1 0 1 1 1 1
- 47. Solution Find A[n] for all positive integers n. 0 0 1 A = 1 0 0 1 1 0 1 1 1 A[4] = A[3] ⨀ A 1 0 1 1 1 1 1 1 1 A[5] = A[4] ⨀ A 1 1 1 1 1 1
- 48. Solution Find A[n] for all positive integers n. Therefore, A[n] = A[5] for all positive integers n with n ≥ 5. 0 0 1 A = 1 0 0 1 1 0 1 1 1 A[4] = A[3] ⨀ A 1 0 1 1 1 1 1 1 1 A[5] = A[4] ⨀ A 1 1 1 1 1 1
- 49. Any Question?
- 50. oSet November 12 – December 05 50 Reminders
- 51. EXAM RESULT
- 52. 86.96% 83.04% 65.00% 27.83% 53.04% 86.23% 45.65% 41.85% 56.39% 74.78% 81.30% 53.04% 75.22% I-1 I-2 I-3 I-4 I-5 I-6 II-1 II-2 II-3 II-4 II-5 II-6 II-7 Best Student Performance with only 13.04% FAILURE 38 39 22 3 23 43 13 18 28 41 41 24 38 I-1 I-2 I-3 I-4 I-5 I-6 II-1 II-2 II-3 II-4 II-5 II-6 II-7 Passed,… Failed, 13.04% Lowest Student Performance with almost 72.17% FAILURE CMSC 56 3rd Long Exam | AY 2018 –2019 06Students failed EXAM PARTSHistogram Students passed in Part I-6 40Students passed Students failed in Part I- 4 43 43 PARTI-1 PARTI-4
- 53. 100% 105.50 Samson, Aron Miles 100% 103.50 Garcia, Patricia Marie 100% 102.00 Ngo, Ma. Gishelle Anne 97.5% 97.50 Gavieta, Don Michael 96.0% 96.00 Pigason, Jasper Robert CMSC 56 3rd LE Top 5 Scorers Overall

- With matrices, we set the diagonal to all 1’s