1 of 53

What's hot

Basic Concept Of Probability
Basic Concept Of Probabilityguest45a926

Matrices and determinants
Matrices and determinantsKum Visal

CMSC 56 | Lecture 6: Sets & Set Operations
CMSC 56 | Lecture 6: Sets & Set Operationsallyn joy calcaben

PROBABILITY
PROBABILITYVIV13

Introduction to Graph Theory
Introduction to Graph Theory Kazi Md. Saidul

Permutation & Combination
Permutation & CombinationPuru Agrawal

Mathematical induction
Mathematical inductionSman Abbasi

Introduction of Probability
Introduction of Probabilityrey castro

Learn Set Theory
Learn Set Theoryyochevedl

Matrix multiplication, inverse
Matrix multiplication, inversePrasanth George

Determinants - Mathematics
Determinants - MathematicsDrishti Bhalla

Lesson 1.2 the set of real numbers
Lesson 1.2 the set of real numbersJohnnyBallecer

10.2 using combinations and the binomial theorem
10.2 using combinations and the binomial theoremhartcher

01_Probability of Simple Events.ppt
01_Probability of Simple Events.pptReinabelleMarquez1

Matrix and its operation (addition, subtraction, multiplication)
Matrix and its operation (addition, subtraction, multiplication)NirnayMukharjee

What's hot(20)

Basic Concept Of Probability
Basic Concept Of Probability

Matrices and determinants
Matrices and determinants

Matrix algebra
Matrix algebra

CMSC 56 | Lecture 6: Sets & Set Operations
CMSC 56 | Lecture 6: Sets & Set Operations

PROBABILITY
PROBABILITY

Introduction to Graph Theory
Introduction to Graph Theory

Permutation & Combination
Permutation & Combination

Relations in Discrete Math
Relations in Discrete Math

the inverse of the matrix
the inverse of the matrix

Mathematical induction
Mathematical induction

Introduction of Probability
Introduction of Probability

Matrices
Matrices

Learn Set Theory
Learn Set Theory

Matrix multiplication, inverse
Matrix multiplication, inverse

Determinants - Mathematics
Determinants - Mathematics

Lesson 1.2 the set of real numbers
Lesson 1.2 the set of real numbers

Logarithms
Logarithms

10.2 using combinations and the binomial theorem
10.2 using combinations and the binomial theorem

01_Probability of Simple Events.ppt
01_Probability of Simple Events.ppt

Matrix and its operation (addition, subtraction, multiplication)
Matrix and its operation (addition, subtraction, multiplication)

Similar to CMSC 56 | Lecture 17: Matrices

Similar to CMSC 56 | Lecture 17: Matrices(20)

Bba i-bm-u-2- matrix -
Bba i-bm-u-2- matrix -

chap01987654etghujh76687976jgtfhhhgve.ppt
chap01987654etghujh76687976jgtfhhhgve.ppt

Matrices
Matrices

Matrix and Determinants
Matrix and Determinants

Per5_matrix multiplication
Per5_matrix multiplication

Matrices & determinants
Matrices & determinants

Matrices & Determinants
Matrices & Determinants

Matrices 1.pdf
Matrices 1.pdf

SETS
SETS

Metrix[1]
Metrix[1]

01_Sets.pdf
01_Sets.pdf

presentationonmatrix-160801150449 (1).pptx
presentationonmatrix-160801150449 (1).pptx

Matrices & Determinants.pdf
Matrices & Determinants.pdf

Sets
Sets

Matrices and determinants_01
Matrices and determinants_01

SETS - Vedantu.pdf
SETS - Vedantu.pdf

Takue
Takue

Linear Algebra.pptx
Linear Algebra.pptx

schaums-probability.pdf
schaums-probability.pdf

More from allyn joy calcaben

CMSC 56 | Lecture 16: Equivalence of Relations & Partial Ordering
CMSC 56 | Lecture 16: Equivalence of Relations & Partial Orderingallyn joy calcaben

CMSC 56 | Lecture 15: Closures of Relations
CMSC 56 | Lecture 15: Closures of Relationsallyn joy calcaben

CMSC 56 | Lecture 14: Representing Relations
CMSC 56 | Lecture 14: Representing Relationsallyn joy calcaben

CMSC 56 | Lecture 13: Relations and their Properties
CMSC 56 | Lecture 13: Relations and their Propertiesallyn joy calcaben

CMSC 56 | Lecture 12: Recursive Definition & Algorithms, and Program Correctness
CMSC 56 | Lecture 12: Recursive Definition & Algorithms, and Program Correctnessallyn joy calcaben

CMSC 56 | Lecture 11: Mathematical Induction
CMSC 56 | Lecture 11: Mathematical Inductionallyn joy calcaben

CMSC 56 | Lecture 10: Integer Representations & Algorithms
CMSC 56 | Lecture 10: Integer Representations & Algorithmsallyn joy calcaben

CMSC 56 | Lecture 9: Functions Representations
CMSC 56 | Lecture 9: Functions Representationsallyn joy calcaben

CMSC 56 | Lecture 8: Growth of Functions
CMSC 56 | Lecture 8: Growth of Functionsallyn joy calcaben

CMSC 56 | Lecture 4: Rules of Inference
CMSC 56 | Lecture 4: Rules of Inferenceallyn joy calcaben

CMSC 56 | Lecture 5: Proofs Methods and Strategy
CMSC 56 | Lecture 5: Proofs Methods and Strategyallyn joy calcaben

CMSC 56 | Lecture 3: Predicates & Quantifiers
CMSC 56 | Lecture 3: Predicates & Quantifiersallyn joy calcaben

CMSC 56 | Lecture 2: Propositional Equivalences
CMSC 56 | Lecture 2: Propositional Equivalencesallyn joy calcaben

CMSC 56 | Lecture 1: Propositional Logic
CMSC 56 | Lecture 1: Propositional Logicallyn joy calcaben

La Solidaridad and the Propaganda Movement
La Solidaridad and the Propaganda Movementallyn joy calcaben

Computer Vision: Feature matching with RANSAC Algorithm
Computer Vision: Feature matching with RANSAC Algorithmallyn joy calcaben

Chapter 7 expressions and assignment statements ii
Chapter 7 expressions and assignment statements iiallyn joy calcaben

More from allyn joy calcaben(18)

CMSC 56 | Lecture 16: Equivalence of Relations & Partial Ordering
CMSC 56 | Lecture 16: Equivalence of Relations & Partial Ordering

CMSC 56 | Lecture 15: Closures of Relations
CMSC 56 | Lecture 15: Closures of Relations

CMSC 56 | Lecture 14: Representing Relations
CMSC 56 | Lecture 14: Representing Relations

CMSC 56 | Lecture 13: Relations and their Properties
CMSC 56 | Lecture 13: Relations and their Properties

CMSC 56 | Lecture 12: Recursive Definition & Algorithms, and Program Correctness
CMSC 56 | Lecture 12: Recursive Definition & Algorithms, and Program Correctness

CMSC 56 | Lecture 11: Mathematical Induction
CMSC 56 | Lecture 11: Mathematical Induction

CMSC 56 | Lecture 10: Integer Representations & Algorithms
CMSC 56 | Lecture 10: Integer Representations & Algorithms

CMSC 56 | Lecture 9: Functions Representations
CMSC 56 | Lecture 9: Functions Representations

CMSC 56 | Lecture 8: Growth of Functions
CMSC 56 | Lecture 8: Growth of Functions

CMSC 56 | Lecture 4: Rules of Inference
CMSC 56 | Lecture 4: Rules of Inference

CMSC 56 | Lecture 5: Proofs Methods and Strategy
CMSC 56 | Lecture 5: Proofs Methods and Strategy

CMSC 56 | Lecture 3: Predicates & Quantifiers
CMSC 56 | Lecture 3: Predicates & Quantifiers

CMSC 56 | Lecture 2: Propositional Equivalences
CMSC 56 | Lecture 2: Propositional Equivalences

CMSC 56 | Lecture 1: Propositional Logic
CMSC 56 | Lecture 1: Propositional Logic

La Solidaridad and the Propaganda Movement
La Solidaridad and the Propaganda Movement

Computer Vision: Feature matching with RANSAC Algorithm
Computer Vision: Feature matching with RANSAC Algorithm

Chapter 7 expressions and assignment statements ii
Chapter 7 expressions and assignment statements ii

#11 osteoporosis
#11 osteoporosis

Personalisation of Education by AI and Big Data - Lourdes Guàrdia
Personalisation of Education by AI and Big Data - Lourdes GuàrdiaEADTU

Model Attribute _rec_name in the Odoo 17
Model Attribute _rec_name in the Odoo 17Celine George

MuleSoft Integration with AWS Textract | Calling AWS Textract API |AWS - Clou...
MuleSoft Integration with AWS Textract | Calling AWS Textract API |AWS - Clou...MysoreMuleSoftMeetup

Unit 3 Emotional Intelligence and Spiritual Intelligence.pdf
Unit 3 Emotional Intelligence and Spiritual Intelligence.pdfDr Vijay Vishwakarma

Simple, Complex, and Compound Sentences Exercises.pdf
Simple, Complex, and Compound Sentences Exercises.pdfstareducators107

When Quality Assurance Meets Innovation in Higher Education - Report launch w...
When Quality Assurance Meets Innovation in Higher Education - Report launch w...Gary Wood

QUATER-1-PE-HEALTH-LC2- this is just a sample of unpacked lesson
QUATER-1-PE-HEALTH-LC2- this is just a sample of unpacked lessonhttgc7rh9c

What is 3 Way Matching Process in Odoo 17.pptx
What is 3 Way Matching Process in Odoo 17.pptxCeline George

SPLICE Working Group: Reusable Code Examples
SPLICE Working Group: Reusable Code ExamplesPeter Brusilovsky

How to Add a Tool Tip to a Field in Odoo 17
How to Add a Tool Tip to a Field in Odoo 17Celine George

80 ĐỀ THI THỬ TUYỂN SINH TIẾNG ANH VÀO 10 SỞ GD – ĐT THÀNH PHỐ HỒ CHÍ MINH NĂ...
80 ĐỀ THI THỬ TUYỂN SINH TIẾNG ANH VÀO 10 SỞ GD – ĐT THÀNH PHỐ HỒ CHÍ MINH NĂ...Nguyen Thanh Tu Collection

UGC NET Paper 1 Unit 7 DATA INTERPRETATION.pdf
UGC NET Paper 1 Unit 7 DATA INTERPRETATION.pdfNirmal Dwivedi

Graduate Outcomes Presentation Slides - English
Graduate Outcomes Presentation Slides - Englishneillewis46

HMCS Vancouver Pre-Deployment Brief - May 2024 (Web Version).pptx
HMCS Vancouver Pre-Deployment Brief - May 2024 (Web Version).pptxmarlenawright1

dusjagr & nano talk on open tools for agriculture research and learning
dusjagr & nano talk on open tools for agriculture research and learningMarc Dusseiller Dusjagr

Understanding Accommodations and Modifications
Understanding Accommodations and ModificationsMJDuyan

NO1 Top Black Magic Specialist In Lahore Black magic In Pakistan Kala Ilam Ex...
NO1 Top Black Magic Specialist In Lahore Black magic In Pakistan Kala Ilam Ex...Amil baba

Diuretic, Hypoglycemic and Limit test of Heavy metals and Arsenic.-1.pdf
Diuretic, Hypoglycemic and Limit test of Heavy metals and Arsenic.-1.pdfKartik Tiwari

Personalisation of Education by AI and Big Data - Lourdes Guàrdia
Personalisation of Education by AI and Big Data - Lourdes Guàrdia

Model Attribute _rec_name in the Odoo 17
Model Attribute _rec_name in the Odoo 17

MuleSoft Integration with AWS Textract | Calling AWS Textract API |AWS - Clou...
MuleSoft Integration with AWS Textract | Calling AWS Textract API |AWS - Clou...

Unit 3 Emotional Intelligence and Spiritual Intelligence.pdf
Unit 3 Emotional Intelligence and Spiritual Intelligence.pdf

VAMOS CUIDAR DO NOSSO PLANETA! .
VAMOS CUIDAR DO NOSSO PLANETA! .

Simple, Complex, and Compound Sentences Exercises.pdf
Simple, Complex, and Compound Sentences Exercises.pdf

When Quality Assurance Meets Innovation in Higher Education - Report launch w...
When Quality Assurance Meets Innovation in Higher Education - Report launch w...

QUATER-1-PE-HEALTH-LC2- this is just a sample of unpacked lesson
QUATER-1-PE-HEALTH-LC2- this is just a sample of unpacked lesson

What is 3 Way Matching Process in Odoo 17.pptx
What is 3 Way Matching Process in Odoo 17.pptx

SPLICE Working Group: Reusable Code Examples
SPLICE Working Group: Reusable Code Examples

How to Add a Tool Tip to a Field in Odoo 17
How to Add a Tool Tip to a Field in Odoo 17

80 ĐỀ THI THỬ TUYỂN SINH TIẾNG ANH VÀO 10 SỞ GD – ĐT THÀNH PHỐ HỒ CHÍ MINH NĂ...
80 ĐỀ THI THỬ TUYỂN SINH TIẾNG ANH VÀO 10 SỞ GD – ĐT THÀNH PHỐ HỒ CHÍ MINH NĂ...

UGC NET Paper 1 Unit 7 DATA INTERPRETATION.pdf
UGC NET Paper 1 Unit 7 DATA INTERPRETATION.pdf

Graduate Outcomes Presentation Slides - English
Graduate Outcomes Presentation Slides - English

HMCS Vancouver Pre-Deployment Brief - May 2024 (Web Version).pptx
HMCS Vancouver Pre-Deployment Brief - May 2024 (Web Version).pptx

Including Mental Health Support in Project Delivery, 14 May.pdf
Including Mental Health Support in Project Delivery, 14 May.pdf

dusjagr & nano talk on open tools for agriculture research and learning
dusjagr & nano talk on open tools for agriculture research and learning

Understanding Accommodations and Modifications
Understanding Accommodations and Modifications

NO1 Top Black Magic Specialist In Lahore Black magic In Pakistan Kala Ilam Ex...
NO1 Top Black Magic Specialist In Lahore Black magic In Pakistan Kala Ilam Ex...

Diuretic, Hypoglycemic and Limit test of Heavy metals and Arsenic.-1.pdf
Diuretic, Hypoglycemic and Limit test of Heavy metals and Arsenic.-1.pdf

CMSC 56 | Lecture 17: Matrices

• 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
• 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
• 50. oSet November 12 – December 05 50 Reminders
• 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

Editor's Notes

1. With matrices, we set the diagonal to all 1’s
Current LanguageEnglish
Español
Portugues
Français
Deutsche