The document discusses matrices and determinants. It defines different types of matrices like rectangular, square, diagonal, scalar, row, column, identity and zero matrices. It explains how to find the determinant of matrices of order 1, 2 and 3 by expansion along the first row. It also defines minors, cofactors and properties of determinants. It describes how to perform row and column operations to evaluate determinants.
The document discusses matrices and determinants. It defines different types of matrices like rectangular, square, diagonal, scalar, row, column, identity and zero matrices. It explains how to find the determinant of matrices of order 1, 2 and 3 by expansion along the first row. It also defines minors, cofactors and properties of determinants. It describes how to perform row and column operations to evaluate determinants. Finally, it provides examples to calculate determinants.
1. The document discusses matrices and determinants, including types of matrices like rectangular, square, diagonal, and scalar matrices.
2. It defines determinants and provides rules for computing determinants of matrices of order 2 and 3 by expanding along rows or columns.
3. Key concepts covered include minors, cofactors, properties of determinants like how row operations affect the determinant value, and examples of computing determinants.
1. The document discusses matrices and determinants. It defines different types of matrices such as rectangular, square, diagonal, scalar, row, column, identity, zero, upper triangular, and lower triangular matrices.
2. It explains how to calculate determinants of matrices. The determinant of a 1x1 matrix is the single element. The determinant of a 2x2 matrix is calculated using a formula. Determinants of higher order matrices are calculated by expanding along rows or columns.
3. It introduces concepts of minors, cofactors, and explains how the value of a determinant can be written in terms of its minors and cofactors. It also lists some properties and operations for determinants.
Determinants provide a scalar quantity associated with square matrices. There are several properties of determinants, including that the determinant of a matrix does not change if rows or columns are interchanged. The determinant can be expressed as the sum of the products of each element and its corresponding cofactor. Examples show how to evaluate determinants by expanding along rows or columns and applying properties such as identical rows resulting in a determinant of zero.
Determinants provide a scalar quantity associated with square matrices. There are several properties of determinants, including that the determinant of a matrix does not change if rows or columns are interchanged. The determinant can be expressed as the sum of the products of each element and its corresponding cofactor. Examples show how to evaluate determinants by expanding along rows or columns and applying properties such as identical rows resulting in a determinant of zero.
- The document discusses determinants of square matrices, including how to calculate the determinant of matrices of various orders, properties of determinants, and some applications of determinants.
- Key concepts covered include minors, cofactors, expanding determinants in terms of minors and cofactors, properties such as how determinants change with row/column operations, and using determinants to solve systems of linear equations.
- Examples are provided to demonstrate calculating determinants and using properties to simplify or prove identities about determinants.
This document provides information about determinants of square matrices:
- It defines the determinant of a matrix as a scalar value associated with the matrix. Determinants are computed using minors and cofactors.
- Properties of determinants are described, such as how determinants change with row/column operations or identical rows/columns.
- Examples are provided to demonstrate computing determinants by expanding along rows or columns and using cofactors and minors.
- Applications of determinants include finding the area of triangles and solving systems of linear equations.
The document is a maths project report for class 12th student Tabrez Khan on the topic of determinants. It contains definitions and properties of determinants of order 1, 2 and 3 matrices. It discusses minors, cofactors and applications of determinants like solving systems of linear equations using Cramer's rule. It also contains examples of evaluating determinants and applying properties of determinants to simplify expressions.
The document discusses matrices and determinants. It defines different types of matrices like rectangular, square, diagonal, scalar, row, column, identity and zero matrices. It explains how to find the determinant of matrices of order 1, 2 and 3 by expansion along the first row. It also defines minors, cofactors and properties of determinants. It describes how to perform row and column operations to evaluate determinants. Finally, it provides examples to calculate determinants.
1. The document discusses matrices and determinants, including types of matrices like rectangular, square, diagonal, and scalar matrices.
2. It defines determinants and provides rules for computing determinants of matrices of order 2 and 3 by expanding along rows or columns.
3. Key concepts covered include minors, cofactors, properties of determinants like how row operations affect the determinant value, and examples of computing determinants.
1. The document discusses matrices and determinants. It defines different types of matrices such as rectangular, square, diagonal, scalar, row, column, identity, zero, upper triangular, and lower triangular matrices.
2. It explains how to calculate determinants of matrices. The determinant of a 1x1 matrix is the single element. The determinant of a 2x2 matrix is calculated using a formula. Determinants of higher order matrices are calculated by expanding along rows or columns.
3. It introduces concepts of minors, cofactors, and explains how the value of a determinant can be written in terms of its minors and cofactors. It also lists some properties and operations for determinants.
Determinants provide a scalar quantity associated with square matrices. There are several properties of determinants, including that the determinant of a matrix does not change if rows or columns are interchanged. The determinant can be expressed as the sum of the products of each element and its corresponding cofactor. Examples show how to evaluate determinants by expanding along rows or columns and applying properties such as identical rows resulting in a determinant of zero.
Determinants provide a scalar quantity associated with square matrices. There are several properties of determinants, including that the determinant of a matrix does not change if rows or columns are interchanged. The determinant can be expressed as the sum of the products of each element and its corresponding cofactor. Examples show how to evaluate determinants by expanding along rows or columns and applying properties such as identical rows resulting in a determinant of zero.
- The document discusses determinants of square matrices, including how to calculate the determinant of matrices of various orders, properties of determinants, and some applications of determinants.
- Key concepts covered include minors, cofactors, expanding determinants in terms of minors and cofactors, properties such as how determinants change with row/column operations, and using determinants to solve systems of linear equations.
- Examples are provided to demonstrate calculating determinants and using properties to simplify or prove identities about determinants.
This document provides information about determinants of square matrices:
- It defines the determinant of a matrix as a scalar value associated with the matrix. Determinants are computed using minors and cofactors.
- Properties of determinants are described, such as how determinants change with row/column operations or identical rows/columns.
- Examples are provided to demonstrate computing determinants by expanding along rows or columns and using cofactors and minors.
- Applications of determinants include finding the area of triangles and solving systems of linear equations.
The document is a maths project report for class 12th student Tabrez Khan on the topic of determinants. It contains definitions and properties of determinants of order 1, 2 and 3 matrices. It discusses minors, cofactors and applications of determinants like solving systems of linear equations using Cramer's rule. It also contains examples of evaluating determinants and applying properties of determinants to simplify expressions.
This document is the preface to the instructor's manual for Classical Dynamics of Particles and Systems by Stephen T. Thornton and Jerry B. Marion. It provides an overview of the contents of the manual, which contains solutions to the end-of-chapter problems from the textbook. The preface notes there are now 509 problems and the solutions range from straightforward to challenging. It stresses the solutions are only for instructors and should not be shared with students.
Chapter 3: Linear Systems and Matrices - Part 3/SlidesChaimae Baroudi
The document discusses determinants of matrices. Some key points:
- The determinant (det) of a square matrix is a single number that can be used to determine properties of the matrix, such as invertibility.
- Formulas are given for calculating the determinant of matrices based on their size, such as the cofactor expansion method.
- Certain types of matrices have simple determinant values, such as triangular and diagonal matrices. The determinant of a triangular matrix is the product of its diagonal entries, and the determinant of a diagonal matrix is the product of its diagonal entries.
This document provides an overview of matrix methods for solving systems of linear equations. It begins with an example from structural engineering of setting up a system of 6 equations with 6 unknowns to model the forces and reactions in a statically determinant truss. The equations are represented in matrix notation as [A]{x}={c}. The document then reviews key matrix concepts and operations used to solve systems of linear equations, such as matrix addition, multiplication, transposes, inverses, and types of matrices. It aims to help readers understand how to set up and solve systems of linear equations using matrices.
MATRICES maths project.pptxsgdhdghdgf gr to f HR fpremkumar24914
Dfgvvvgggggggdcdfggsggshhdbdybfhfbfhbfbfhfbfhbfbfhfhfjfhfhhfhfhgththththhththththththhhfjfbrhhrbtht me for the information ℹ️ℹ️ and contact information ℹ️ℹ️ and grant you are you 💕💕💕 I don't have main roll no data for the same and the information about it and grant you are not coming to the parent or no data and contact information about the information of the elite year 2 years blood group of the elite year 2 years of the elite of nobody will give me when the parent is not 🚫 to come in detail and t be a consideration and grant you the parent or not feeling well as well as the elite of the parent iam in the parent iam in the parent iam not 🚭🚭 I will kick u in the shadow and the information about the information of the elite and the information of nobody in a sec and grant me a call 🤙🤙 and grant you are you still he is a consideration of my life 🧬🧬 I don't have main places to visit in the shadow of the elite year 2 years of life and the information about it but na vit d I don't have any other questions for you are not coming school year blood test 😔 😔 for jee mains ka I will be taking leave for the information of nobody will give you a sec i will be able and grant me a day or no data for the information about it but I don't have main places to be taking leave for jee to be able to be a sec i will confirm with you are not coming school year 2 days I will be able and grant you to talk to you and contact number of the parent is a consideration and the information about you to be taking the information of nobody will give me when I am in a sec to you and grant you to talk about the parent is not coming to talk about it and I am in a consideration of the day of life and grant me leave taken to talk to you sir and grant me a call 🤙 I am in a sec to you and grant you are not coming school year blood test postion and grant you are not coming school year blood test postion and grant you are not coming school year blood test postion and grant you are not coming school year blood test postion and grant you are not coming school year blood test postion and grant you are not coming school year blood test postion and grant you are not coming school year blood test and grant you are not coming school today my future is not coming to school today my future life and grant me leave taken to the parent or no data for the information of nobody will give me a sec to you sir and the parent iam waiting on the parent iam in detail and grant you are not coming school year 2 years blood test postion and contact information about the information of nobody is okay va va va va va for the information of your work and grant you the same and grant you the same and grant me a call me you will be able to attend a sec to you sir for your work is not coming school year blood test postion and grant you the same to talk to you sir for the same yyrty and grant you are not coming school year blood test postion for the same yyrty and grant you are not available for jee class
Determinants, crammers law, Inverse by adjoint and the applicationsNikoBellic28
The document discusses various topics related to matrices including determinants, Cramer's rule, and applications of matrices. It provides definitions and examples of determinants, properties of determinants, calculating a 2x2 determinant, and Cramer's rule for 2x2 and 3x3 matrices. It also demonstrates finding the inverse of a matrix using the adjoint method and provides an example of using matrices to solve a system of linear equations.
Matrices can be added, subtracted, and multiplied according to certain rules.
- Matrices can only be added or subtracted if they are the same size. The sum or difference of matrices A and B yields a matrix C of the same size.
- Matrices can be multiplied by a scalar. Multiplying a matrix A by a scalar k results in a new matrix kA where each element is multiplied by k.
- Matrix multiplication allows combining information from two matrices but has specific rules regarding the dimensions of the matrices.
This document discusses the inverse of matrices. It defines the cofactor method for finding the inverse of a matrix, which involves calculating the matrix of cofactors and then taking its transpose divided by the determinant of the original matrix. Several examples are worked through, including calculating the inverse of a 3x3 matrix. The document also discusses using matrices to represent and solve systems of simultaneous linear equations, developing the general matrix solution of x = A^-1b where A is the coefficient matrix, x is the vector of unknowns, and b is the vector of constants.
The document summarizes properties and techniques for computing determinants of matrices. It defines determinants and provides examples of computing 2x2 and 3x3 determinants. It discusses properties of determinants including how determinants change when multiplying a matrix by a scalar or performing row operations. It also defines what makes a matrix singular versus nonsingular.
This document defines determinants and provides examples of calculating determinants of 2x2 and 3x3 matrices. It introduces key terminology like minors, cofactors, and expands on Cramer's rule for solving systems of linear equations using determinants. The document explains that the determinant of a square matrix is a single number associated with the matrix, and provides rules and examples for calculating determinants by summing the products of elements and their cofactors.
This presentation describes Matrices and Determinants in detail including all the relevant definitions with examples, various concepts and the practice problems.
The document provides information about a test for candidates applying for M.Tech in Computer Science. It consists of two parts - Test MIII in the morning and Test CS in the afternoon. Test CS has two groups - Group A containing questions on analytical ability and mathematics, and Group B containing subject-specific questions in one of several sections according to the candidate's choice. The document then provides sample questions for Group A (mathematics-based) and Group B (subject-specific for various domains like mathematics, statistics, physics, computer science, and engineering).
The document provides information about a test for candidates applying to an M.Tech. program in Computer Science. [The test consists of two parts - an objective test in the morning and a short answer test in the afternoon. The short answer test has two groups - Group A covers analytical ability and mathematics at the B.Sc. pass level, and Group B covers various subjects at higher levels and the candidate must choose one section to answer questions from according to their background.] The document then provides sample questions that may be asked in Group A, which covers topics in mathematics including calculus, linear algebra, number theory, and sequences and series.
This document provides an introduction to matrices. It defines a matrix as a rectangular array of numbers or other items arranged in rows and columns. Matrices are conventionally sized using the number of rows and columns. The document outlines basic matrix operations such as addition, subtraction, scalar multiplication, and matrix multiplication. It also defines key matrix types including identity, diagonal, triangular, and transpose matrices.
ALLIED MATHEMATICS -I UNIT III MATRICES.pptssuser2e348b
Matrices can be represented as arrays of numbers arranged in rows and columns. A matrix is defined by its dimensions (number of rows and columns). There are several types of matrices including square, rectangular, diagonal, identity, null, triangular, and scalar matrices. Operations on matrices include addition, subtraction, and multiplication. For matrices to be added or subtracted, they must be the same size. Matrices can be multiplied by a scalar value. Matrix multiplication results in another matrix, and the number of columns of the first matrix must equal the number of rows of the second matrix.
This document appears to be an assignment submission for a financial engineering course. It includes a plagiarism declaration signed by the student, Andrew Hair. The assignment contains 11 questions addressing interest rate derivatives and modeling using the Vasicek model. Code is provided in MATLAB to generate simulations and analyze interest rate data based on the questions.
This document provides an overview of the topics covered in an introductory mathematics analysis course for business, economics, and social sciences. It includes:
1) A review of key concepts like algebra, subsets of real numbers, properties of operations, and graphing numbers on a number line.
2) An outline of course structure with sections on algebra, algebraic expressions, fractions, and mathematical systems.
3) Examples of problems and their step-by-step solutions covering topics like simplifying expressions, factoring, addition/subtraction of fractions, and properties of real numbers.
The document discusses determinants of matrices and their geometric interpretations. It begins by defining a matrix as a rectangular table of numbers or formulas. It then explains that the determinant of a 2x2 matrix gives the signed area of the parallelogram defined by the row vectors of the matrix. The sign of the determinant indicates whether the parallelogram is swept clockwise or counterclockwise. Finally, it generalizes these concepts to 3x3 matrices by writing out the formula for the determinant as a sum of signed areas of parallelograms.
K-Notes are concise study materials intended for quick revision near the end of preparation for exams like GATE. Each K-Note covers the concepts from a subject in 40 pages or less. They are useful for final preparation and travel. Students should use K-Notes in the last 2 months before the exam, practicing questions after reviewing each note. The document then provides a summary of key concepts in linear algebra and matrices, including matrix properties, operations, inverses, and systems of linear equations.
This document is the preface to the instructor's manual for Classical Dynamics of Particles and Systems by Stephen T. Thornton and Jerry B. Marion. It provides an overview of the contents of the manual, which contains solutions to the end-of-chapter problems from the textbook. The preface notes there are now 509 problems and the solutions range from straightforward to challenging. It stresses the solutions are only for instructors and should not be shared with students.
Chapter 3: Linear Systems and Matrices - Part 3/SlidesChaimae Baroudi
The document discusses determinants of matrices. Some key points:
- The determinant (det) of a square matrix is a single number that can be used to determine properties of the matrix, such as invertibility.
- Formulas are given for calculating the determinant of matrices based on their size, such as the cofactor expansion method.
- Certain types of matrices have simple determinant values, such as triangular and diagonal matrices. The determinant of a triangular matrix is the product of its diagonal entries, and the determinant of a diagonal matrix is the product of its diagonal entries.
This document provides an overview of matrix methods for solving systems of linear equations. It begins with an example from structural engineering of setting up a system of 6 equations with 6 unknowns to model the forces and reactions in a statically determinant truss. The equations are represented in matrix notation as [A]{x}={c}. The document then reviews key matrix concepts and operations used to solve systems of linear equations, such as matrix addition, multiplication, transposes, inverses, and types of matrices. It aims to help readers understand how to set up and solve systems of linear equations using matrices.
MATRICES maths project.pptxsgdhdghdgf gr to f HR fpremkumar24914
Dfgvvvgggggggdcdfggsggshhdbdybfhfbfhbfbfhfbfhbfbfhfhfjfhfhhfhfhgththththhththththththhhfjfbrhhrbtht me for the information ℹ️ℹ️ and contact information ℹ️ℹ️ and grant you are you 💕💕💕 I don't have main roll no data for the same and the information about it and grant you are not coming to the parent or no data and contact information about the information of the elite year 2 years blood group of the elite year 2 years of the elite of nobody will give me when the parent is not 🚫 to come in detail and t be a consideration and grant you the parent or not feeling well as well as the elite of the parent iam in the parent iam in the parent iam not 🚭🚭 I will kick u in the shadow and the information about the information of the elite and the information of nobody in a sec and grant me a call 🤙🤙 and grant you are you still he is a consideration of my life 🧬🧬 I don't have main places to visit in the shadow of the elite year 2 years of life and the information about it but na vit d I don't have any other questions for you are not coming school year blood test 😔 😔 for jee mains ka I will be taking leave for the information of nobody will give you a sec i will be able and grant me a day or no data for the information about it but I don't have main places to be taking leave for jee to be able to be a sec i will confirm with you are not coming school year 2 days I will be able and grant you to talk to you and contact number of the parent is a consideration and the information about you to be taking the information of nobody will give me when I am in a sec to you and grant you to talk about the parent is not coming to talk about it and I am in a consideration of the day of life and grant me leave taken to talk to you sir and grant me a call 🤙 I am in a sec to you and grant you are not coming school year blood test postion and grant you are not coming school year blood test postion and grant you are not coming school year blood test postion and grant you are not coming school year blood test postion and grant you are not coming school year blood test postion and grant you are not coming school year blood test postion and grant you are not coming school year blood test and grant you are not coming school today my future is not coming to school today my future life and grant me leave taken to the parent or no data for the information of nobody will give me a sec to you sir and the parent iam waiting on the parent iam in detail and grant you are not coming school year 2 years blood test postion and contact information about the information of nobody is okay va va va va va for the information of your work and grant you the same and grant you the same and grant me a call me you will be able to attend a sec to you sir for your work is not coming school year blood test postion and grant you the same to talk to you sir for the same yyrty and grant you are not coming school year blood test postion for the same yyrty and grant you are not available for jee class
Determinants, crammers law, Inverse by adjoint and the applicationsNikoBellic28
The document discusses various topics related to matrices including determinants, Cramer's rule, and applications of matrices. It provides definitions and examples of determinants, properties of determinants, calculating a 2x2 determinant, and Cramer's rule for 2x2 and 3x3 matrices. It also demonstrates finding the inverse of a matrix using the adjoint method and provides an example of using matrices to solve a system of linear equations.
Matrices can be added, subtracted, and multiplied according to certain rules.
- Matrices can only be added or subtracted if they are the same size. The sum or difference of matrices A and B yields a matrix C of the same size.
- Matrices can be multiplied by a scalar. Multiplying a matrix A by a scalar k results in a new matrix kA where each element is multiplied by k.
- Matrix multiplication allows combining information from two matrices but has specific rules regarding the dimensions of the matrices.
This document discusses the inverse of matrices. It defines the cofactor method for finding the inverse of a matrix, which involves calculating the matrix of cofactors and then taking its transpose divided by the determinant of the original matrix. Several examples are worked through, including calculating the inverse of a 3x3 matrix. The document also discusses using matrices to represent and solve systems of simultaneous linear equations, developing the general matrix solution of x = A^-1b where A is the coefficient matrix, x is the vector of unknowns, and b is the vector of constants.
The document summarizes properties and techniques for computing determinants of matrices. It defines determinants and provides examples of computing 2x2 and 3x3 determinants. It discusses properties of determinants including how determinants change when multiplying a matrix by a scalar or performing row operations. It also defines what makes a matrix singular versus nonsingular.
This document defines determinants and provides examples of calculating determinants of 2x2 and 3x3 matrices. It introduces key terminology like minors, cofactors, and expands on Cramer's rule for solving systems of linear equations using determinants. The document explains that the determinant of a square matrix is a single number associated with the matrix, and provides rules and examples for calculating determinants by summing the products of elements and their cofactors.
This presentation describes Matrices and Determinants in detail including all the relevant definitions with examples, various concepts and the practice problems.
The document provides information about a test for candidates applying for M.Tech in Computer Science. It consists of two parts - Test MIII in the morning and Test CS in the afternoon. Test CS has two groups - Group A containing questions on analytical ability and mathematics, and Group B containing subject-specific questions in one of several sections according to the candidate's choice. The document then provides sample questions for Group A (mathematics-based) and Group B (subject-specific for various domains like mathematics, statistics, physics, computer science, and engineering).
The document provides information about a test for candidates applying to an M.Tech. program in Computer Science. [The test consists of two parts - an objective test in the morning and a short answer test in the afternoon. The short answer test has two groups - Group A covers analytical ability and mathematics at the B.Sc. pass level, and Group B covers various subjects at higher levels and the candidate must choose one section to answer questions from according to their background.] The document then provides sample questions that may be asked in Group A, which covers topics in mathematics including calculus, linear algebra, number theory, and sequences and series.
This document provides an introduction to matrices. It defines a matrix as a rectangular array of numbers or other items arranged in rows and columns. Matrices are conventionally sized using the number of rows and columns. The document outlines basic matrix operations such as addition, subtraction, scalar multiplication, and matrix multiplication. It also defines key matrix types including identity, diagonal, triangular, and transpose matrices.
ALLIED MATHEMATICS -I UNIT III MATRICES.pptssuser2e348b
Matrices can be represented as arrays of numbers arranged in rows and columns. A matrix is defined by its dimensions (number of rows and columns). There are several types of matrices including square, rectangular, diagonal, identity, null, triangular, and scalar matrices. Operations on matrices include addition, subtraction, and multiplication. For matrices to be added or subtracted, they must be the same size. Matrices can be multiplied by a scalar value. Matrix multiplication results in another matrix, and the number of columns of the first matrix must equal the number of rows of the second matrix.
This document appears to be an assignment submission for a financial engineering course. It includes a plagiarism declaration signed by the student, Andrew Hair. The assignment contains 11 questions addressing interest rate derivatives and modeling using the Vasicek model. Code is provided in MATLAB to generate simulations and analyze interest rate data based on the questions.
This document provides an overview of the topics covered in an introductory mathematics analysis course for business, economics, and social sciences. It includes:
1) A review of key concepts like algebra, subsets of real numbers, properties of operations, and graphing numbers on a number line.
2) An outline of course structure with sections on algebra, algebraic expressions, fractions, and mathematical systems.
3) Examples of problems and their step-by-step solutions covering topics like simplifying expressions, factoring, addition/subtraction of fractions, and properties of real numbers.
The document discusses determinants of matrices and their geometric interpretations. It begins by defining a matrix as a rectangular table of numbers or formulas. It then explains that the determinant of a 2x2 matrix gives the signed area of the parallelogram defined by the row vectors of the matrix. The sign of the determinant indicates whether the parallelogram is swept clockwise or counterclockwise. Finally, it generalizes these concepts to 3x3 matrices by writing out the formula for the determinant as a sum of signed areas of parallelograms.
K-Notes are concise study materials intended for quick revision near the end of preparation for exams like GATE. Each K-Note covers the concepts from a subject in 40 pages or less. They are useful for final preparation and travel. Students should use K-Notes in the last 2 months before the exam, practicing questions after reviewing each note. The document then provides a summary of key concepts in linear algebra and matrices, including matrix properties, operations, inverses, and systems of linear equations.
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
Executive Directors Chat Leveraging AI for Diversity, Equity, and InclusionTechSoup
Let’s explore the intersection of technology and equity in the final session of our DEI series. Discover how AI tools, like ChatGPT, can be used to support and enhance your nonprofit's DEI initiatives. Participants will gain insights into practical AI applications and get tips for leveraging technology to advance their DEI goals.
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Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
Main Java[All of the Base Concepts}.docxadhitya5119
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
The simplified electron and muon model, Oscillating Spacetime: The Foundation...RitikBhardwaj56
Discover the Simplified Electron and Muon Model: A New Wave-Based Approach to Understanding Particles delves into a groundbreaking theory that presents electrons and muons as rotating soliton waves within oscillating spacetime. Geared towards students, researchers, and science buffs, this book breaks down complex ideas into simple explanations. It covers topics such as electron waves, temporal dynamics, and the implications of this model on particle physics. With clear illustrations and easy-to-follow explanations, readers will gain a new outlook on the universe's fundamental nature.
How to Setup Warehouse & Location in Odoo 17 InventoryCeline George
In this slide, we'll explore how to set up warehouses and locations in Odoo 17 Inventory. This will help us manage our stock effectively, track inventory levels, and streamline warehouse operations.
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In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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2. Matrices & Determinants
Session Objectives
• Meaning of matrix
• Type of matrices
• Transpose of Matrix
• Meaning of symmetric and skew symmetric
matrices
• Minor & co-factors
• Computation of adjoint and inverse of a
matrix
3. Matrices & Determinants
TYPES OF MATRICES
NAME DESCRIPTION EXAMPLE
Rectangular
matrix
No. of rows is not equal to
no. of columns
Square matrix No. of rows is equal to no. of
columns
Diagonal
matrix
Non-zero element in principal
diagonal and zero in all other
positions
Scalar matrix Diagonal matrix in which all
the elements on principal
diagonal and same
5
0
2
1
2
6
2 1 3
2 0 1
1 2 4
7
0
0
0
4
0
0
0
2
4
0
0
0
4
0
0
0
4
4. Matrices & Determinants
TYPES OF MATRICES
NAME DESCRIPTION EXAMPLE
Row matrix A matrix with only 1
row
Column matrix A matrix with only I
column
Identity matrix Diagonal matrix
having each
diagonal element
equal to one (I)
Zero matrix A matrix with all zero
entries
3 2 14
2
3
1
0
0
1
0 0
0 0
5. Matrices & Determinants
TYPES OF MATRICES
NAME DESCRIPTION EXAMPLE
Upper Triangular
matrix
Square matrix
having all the entries
zero below the
principal diagonal
Lower Triangular
matrix
Square matrix
having all the entries
zero above the
principal diagonal
7
0
0
6
4
0
3
5
2
7
3
6
0
4
5
0
0
2
6. Matrices & Determinants
Determinants
If is a square matrix of order 1,
then |A| = | a11 | = a11
ij
A = a
If is a square matrix of order 2, then
11 12
21 22
a a
A =
a a
|A| = = a11a22 – a21a12
a a
a a
1
1 1
2
2
1 2
2
8. Matrices & Determinants
Solution
If A = is a square matrix of order 3, then
11 12 13
21 22 23
31 32 33
a a a
a a a
a a a
[Expanding along first row]
11 12 13
22 23 21 23 21 22
21 22 23 11 12 13
32 33 31 33 31 32
31 32 33
a a a
a a a a a a
| A |= a a a = a - a + a
a a a a a a
a a a
11 22 33 32 23 12 21 33 31 23 13 21 32 31 22
= a a a - a a - a a a - a a + a a a - a a
11 22 33 12 31 23 13 21 32 11 23 32 12 21 33 13 31 22
a a a a a a a a a a a a a a a a a a
10. Matrices & Determinants
Minors
-1 4
If A = , then
2 3
21 21 22 22
M = Minor of a = 4, M = Minor of a = -1
11 11 12 12
M = Minor of a = 3, M = Minor of a = 2
11. Matrices & Determinants
Minors
4 7 8
If A = -9 0 0 , then
2 3 4
M11 = Minor of a11 = determinant of the order 2 × 2 square
sub-matrix is obtained by leaving first
row and first column of A
0 0
= = 0
3 4
Similarly, M23 = Minor of a23
4 7
= =12-14=-2
2 3
M32 = Minor of a32 etc.
4 8
= = 0+72 = 72
-9 0
14. Matrices & Determinants
Value of Determinant in Terms
of Minors and Cofactors
11 12 13
21 22 23
31 32 33
a a a
If A = a a a , then
a a a
3 3
i j
ij ij ij ij
j 1 j 1
A 1 a M a C
i1 i1 i2 i2 i3 i3
= a C +a C +a C , for i =1 or i =2 or i =3
15. Matrices & Determinants
Properties of Determinants
1. The value of a determinant remains unchanged, if its
rows and columns are interchanged.
1 1 1 1 2 3
2 2 2 1 2 3
3 3 3 1 2 3
a b c a a a
a b c = b b b
a b c c c c
i e A A
. . '
2. If any two rows (or columns) of a determinant are interchanged,
then the value of the determinant is changed by minus sign.
1 1 1 2 2 2
2 2 2 1 1 1 2 1
3 3 3 3 3 3
a b c a b c
a b c = - a b c R R
a b c a b c
Applying
16. Matrices & Determinants
Properties (Con.)
3. If all the elements of a row (or column) is multiplied by a
non-zero number k, then the value of the new determinant
is k times the value of the original determinant.
1 1 1 1 1 1
2 2 2 2 2 2
3 3 3 3 3 3
ka kb kc a b c
a b c = k a b c
a b c a b c
which also implies
1 1 1 1 1 1
2 2 2 2 2 2
3 3 3 3 3 3
a b c ma mb mc
1
a b c = a b c
m
a b c a b c
17. Matrices & Determinants
Properties (Con.)
4. If each element of any row (or column) consists of
two or more terms, then the determinant can be
expressed as the sum of two or more determinants.
1 1 1 1 1 1 1 1
2 2 2 2 2 2 2 2
3 3 3 3 3 3 3 3
a +x b c a b c x b c
a +y b c = a b c + y b c
a +z b c a b c z b c
5. The value of a determinant is unchanged, if any row
(or column) is multiplied by a number and then added
to any other row (or column).
1 1 1 1 1 1 1 1
2 2 2 2 2 2 2 2 1 1 2 3
3 3 3 3 3 3 3 3
a b c a +mb - nc b c
a b c = a +mb - nc b c C C + mC -nC
a b c a +mb - nc b c
Applying
18. Matrices & Determinants
Properties (Con.)
6. If any two rows (or columns) of a determinant are
identical, then its value is zero.
2 2 2
3 3 3
0 0 0
a b c = 0
a b c
7. If each element of a row (or column) of a determinant is zero,
then its value is zero.
1 1 1
2 2 2
1 1 1
a b c
a b c = 0
a b c
19. Matrices & Determinants
Properties (Con.)
a 0 0
8 Let A = 0 b 0 be a diagonal matrix, then
0 0 c
a 0 0
= 0 b 0
0 0 c
A abc
20. Matrices & Determinants
Row(Column) Operations
Following are the notations to evaluate a determinant:
Similar notations can be used to denote column
operations by replacing R with C.
(i) Ri to denote ith row
(ii) Ri Rj to denote the interchange of ith and jth
rows.
(iii) Ri Ri + lRj to denote the addition of l times the
elements of jth row to the corresponding elements
of ith row.
(iv) lRi to denote the multiplication of all elements of
ith row by l.
21. Matrices & Determinants
Evaluation of Determinants
If a determinant becomes zero on putting
is the factor of the determinant.
x = , then x -
2
3
x 5 2
For example, if Δ = x 9 4 , then at x =2
x 16 8
, because C1 and C2 are identical at x = 2
Hence, (x – 2) is a factor of determinant .
0
22. Matrices & Determinants
Sign System for Expansion of
Determinant
Sign System for order 2 and order 3 are
given by
+ – +
+ –
, – + –
– +
+ – +
23. Matrices & Determinants
42 1 6 6×7 1 6
i 28 7 4 = 4×7 7 4
14 3 2 2×7 3 2
1
6 1 6
=7 4 7 4 Taking out 7 common from C
2 3 2
Example-1
6 -3 2
2 -1 2
-10 5 2
42 1 6
28 7 4
14 3 2
Find the value of the following determinants
(i) (ii)
Solution :
1 3
= 7 × 0 C and C are identical
= 0
24. Matrices & Determinants
Example –1 (ii)
6 -3 2
2 -1 2
-10 5 2
(ii)
3 2 3 2
1 2 1 2
5 2 5 2
1
1 2
3 3 2
( 2) 1 1 2 Taking out 2 common from C
5 5 2
( 2) 0 C and C are identical
0
25. Matrices & Determinants
Evaluate the determinant
1 a b+c
1 b c+a
1 c a+b
Solution :
3 2 3
1 a b+c 1 a a+b+c
1 b c+a = 1 b a+b+c Applying c c +c
1 c a+b 1 c a+b+c
3
1 a 1
= a+b+c 1 b 1 Taking a+b+c common from C
1 c 1
Example - 2
1 3
= a+b + c ×0 C and C are identical
= 0
26. Matrices & Determinants
2 2 2
a b c
We have a b c
bc ca ab
2
1 1 2 2 2 3
(a-b) b-c c
= (a-b)(a+b) (b-c)(b+c) c Applying C C -C and C C -C
-c(a-b) -a(b-c) ab
2
1 2
1 1 c
Taking a-b and b-c common
=(a-b)(b-c) a+b b+c c
from C and C respectively
-c -a ab
Example - 3
bc
2 2 2
a b c
a b c
ca ab
Evaluate the determinant:
Solution:
27. Matrices & Determinants
2
1 1 2
0 1 c
=(a-b)(b-c) -(c-a) b+c c Applying c c -c
-(c-a) -a ab
2
0 1 c
=-(a-b)(b-c)(c-a) 1 b+c c
1 -a ab
2
2 2 3
0 1 c
= -(a-b)(b-c)(c-a) 0 a+b+c c -ab Applying R R -R
1 -a ab
Now expanding along C1 , we get
(a-b) (b-c) (c-a) [- (c2 – ab – ac – bc – c2)]
= (a-b) (b-c) (c-a) (ab + bc + ac)
Solution Cont.
28. Matrices & Determinants
Without expanding the determinant,
prove that
3
3x+y 2x x
4x+3y 3x 3x =x
5x+6y 4x 6x
3x+y 2x x 3x 2x x y 2x x
L.H.S= 4x+3y 3x 3x = 4x 3x 3x + 3y 3x 3x
5x+6y 4x 6x 5x 4x 6x 6y 4x 6x
3 2
3 2 1 1 2 1
= x 4 3 3 +x y 3 3 3
5 4 6 6 4 6
Example-4
Solution :
3 2
1 2
3 2 1
= x 4 3 3 +x y×0 C and C are identical in II determinant
5 4 6
29. Matrices & Determinants
Solution Cont.
3
1 1 2
1 2 1
= x 1 3 3 Applying C C -C
1 4 6
3
2 2 1 3 3 2
1 2 1
=x 0 1 2 ApplyingR R -R and R R -R
0 1 3
3
1
3
= x ×(3-2) Expanding along C
=x = R.H.S.
3
3 2 1
=x 4 3 3
5 4 6
31. Matrices & Determinants
Example-6
2
x+a b c
a x+b c =x (x+a+b+c)
a b x+C
Prove that :
1 1 2 3
x+a b c x+a+b+c b c
L.H.S= a x+b c = x+a+b+c x+b c
a b x+C x+a+b+c b x+c
Applying C C +C +C
Solution :
1
1 b c
= x+a+b+c 1 x+b c
1 b x+c
Taking x+a+b+c commonfrom C
32. Matrices & Determinants
Solution cont.
2 2 1 3 3 1
1 b c
=(x+a+b+c) 0 x 0
0 0 x
Applying R R -R and R R -R
Expanding along C1 , we get
(x + a + b + c) [1(x2)] = x2 (x + a + b + c)
= R.H.S
36. Matrices & Determinants
Solution Cont.
2 2 1 3 3 2
1 2x 2x
=(5x+4) 0 -(x-4) 0 ApplyingR R -R and R R -R
0 x-4 -(x-4)
Now expanding along C1 , we get
2
(5x+4) 1(x- 4) -0
2
=(5x+4)(4-x)
=R.H.S
37. Matrices & Determinants
Example -9
Using properties of determinants, prove that
x+9 x x
x x+9 x =243 (x+3)
x x x+9
x+9 x x
L.H.S= x x+9 x
x x x+9
1 1 2 3
3x+9 x x
= 3x+9 x+9 x Applying C C +C +C
3x+9 x x+9
Solution :
38. Matrices & Determinants
1
=3(x+3) 81 Expanding along C
=243(x+3)
=R.H.S.
1 x x
=(3x+9)1 x+9 x
1 x x+9
Solution Cont.
2 2 1 3 3 2
1 x x
=3 x+3 0 9 0 Applying R R -R and R R -R
0 -9 9
39. Matrices & Determinants
Example -10
Solution :
2 2 2 2 2
2 2 2 2 2
1 1 3
2 2 2 2 2
(b+c) a bc b +c a bc
L.H.S.= (c+a) b ca = c +a b ca Applying C C -2C
(a+b) c ab a +b c ab
2 2 2 2
2 2 2 2
1 1 2
2 2 2 2
a +b +c a bc
a +b +c b ca Applying C C +C
a +b +c c ab
2
2 2 2 2
2
1 a bc
=(a +b +c )1 b ca
1 c ab
2 2
2 2 2 2 2
2 2
(b+c) a bc
(c+a) b ca =(a +b +c )(a-b)(b-c)(c-a)(a+b+c)
(a+b) c ab
Show that
40. Matrices & Determinants
Solution Cont.
2
2 2 2
2 2 1 3 3 2
1 a bc
=(a +b +c ) 0 (b-a)(b+a) c(a-b) Applying R R -R and R R -R
0 (c-b)(c+b) a(b-c)
2 2 2 2 2
1
=(a +b +c )(a-b)(b-c)(-ab-a +bc+c ) Expanding along C
2 2 2
=(a +b +c )(a-b)(b-c)(c-a)(a+b+c)=R.H.S.
2
2 2 2
1 a bc
=(a +b +c )(a-b)(b-c) 0 -(b+a) c
0 -(b+c) a
2 2 2
=(a +b +c )(a-b)(b-c) b c-a + c-a c+a
41. Matrices & Determinants
Applications of Determinants
(Area of a Triangle)
The area of a triangle whose vertices are
is given by the expression
1 1 2 2 3 3
(x , y ), (x , y ) and (x , y )
1 1
2 2
3 3
x y 1
1
Δ= x y 1
2
x y 1
1 2 3 2 3 1 3 1 2
1
= [x (y - y ) + x (y - y ) + x (y - y )]
2
42. Matrices & Determinants
Example
Find the area of a triangle whose
vertices are (-1, 8), (-2, -3) and (3, 2).
Solution :
1 1
2 2
3 3
x y 1 -1 8 1
1 1
Area of triangle= x y 1 = -2 -3 1
2 2
x y 1 3 2 1
1
= -1(-3-2)-8(-2-3)+1(-4+9)
2
1
= 5+40+5 =25 sq.units
2
43. Matrices & Determinants
Condition of Collinearity of
Three Points
If are three points,
then A, B, C are collinear
1 1 2 2 3 3
A (x , y ), B (x , y ) and C (x , y )
1 1 1 1
2 2 2 2
3 3 3 3
Area of triangle ABC =0
x y 1 x y 1
1
x y 1 =0 x y 1 =0
2
x y 1 x y 1
44. Matrices & Determinants
If the points (x, -2) , (5, 2), (8, 8) are collinear,
find x , using determinants.
Example
Solution :
x -2 1
5 2 1 =0
8 8 1
x 2-8 - -2 5-8 +1 40-16 =0
-6x-6+24=0
6x=18 x=3
Since the given points are collinear.
45. Matrices & Determinants
Solution of System of 2 Linear
Equations (Cramer’s Rule)
Let the system of linear equations be
2 2 2
a x+b y = c ... ii
1 1 1
a x+b y = c ... i
1 2
D D
Then x = , y = provided D 0,
D D
1 1 1 1 1 1
1 2
2 2 2 2 2 2
a b c b a c
where D = , D = and D =
a b c b a c
46. Matrices & Determinants
Cramer’s Rule (Con.)
then the system is consistent and has infinitely many
solutions.
1 2
2 If D = 0 and D = D = 0,
then the system is inconsistent and has no solution.
1 If D 0
Note :
,
then the system is consistent and has unique solution.
1 2
3 If D=0 and one of D , D 0,
47. Matrices & Determinants
Example
2 -3
D= =2+9=11 0
3 1
1
7 -3
D = =7+15=22
5 1
2
2 7
D = =10-21=-11
3 5
Solution :
1 2
D 0
D D
22 -11
By Cramer's Rule x= = =2 and y= = =-1
D 11 D 11
Using Cramer's rule , solve the following
system of equations 2x-3y=7, 3x+y=5
48. Matrices & Determinants
Solution of System of 3 Linear
Equations (Cramer’s Rule)
Let the system of linear equations be
2 2 2 2
a x+b y+c z = d ... ii
1 1 1 1
a x+b y+c z = d ... i
3 3 3 3
a x+b y+c z = d ... iii
3
1 2 D
D D
Then x = , y = z = provided D 0,
D D D
,
1 1 1 1 1 1 1 1 1
2 2 2 1 2 2 2 2 2 2 2
3 3 3 3 3 3 3 3 3
a b c d b c a d c
where D = a b c , D = d b c , D = a d c
a b c d b c a d c
1 1 1
3 2 2 2
3 3 3
a b d
and D = a b d
a b d
49. Matrices & Determinants
Cramer’s Rule (Con.)
Note:
(1) If D 0, then the system is consistent and has a unique
solution.
(2) If D=0 and D1 = D2 = D3 = 0, then the system has infinite
solutions or no solution.
(3) If D = 0 and one of D1, D2, D3 0, then the system
is inconsistent and has no solution.
(4) If d1 = d2 = d3 = 0, then the system is called the system of
homogeneous linear equations.
(i) If D 0, then the system has only trivial solution x = y = z = 0.
(ii) If D = 0, then the system has infinite solutions.
51. Matrices & Determinants
3
5 -1 5
D = 2 3 2
5 -2 -1
= 5(-3 +4)+1(-2 - 10)+5(-4-15)
= 5 – 12 – 95 = 5 - 107
= - 102
Solution Cont.
1 2
3
D 0
D D
153 102
By Cramer's Rule x = = =3, y = = =2
D 51 D 51
D -102
and z= = =-2
D 51
2
5 5 4
D = 2 2 5
5 -1 6
= 5(12 +5)+5(12 - 25)+ 4(-2 - 10)
= 85 + 65 – 48 = 150 - 48
= 102
52. Matrices & Determinants
Example
Solve the following system of homogeneous linear equations:
x + y – z = 0, x – 2y + z = 0, 3x + 6y + -5z = 0
Solution:
1 1 - 1
We have D = 1 - 2 1 = 1 10 - 6 - 1 -5 - 3 - 1 6 + 6
3 6 - 5
= 4 + 8 - 12 = 0
The systemhas infinitely many solutions.
Putting z = k, in first two equations, we get
x + y = k, x – 2y = -k
53. Matrices & Determinants
Solution (Con.)
1
k 1
D -k - 2 -2k + k k
By Cramer's rule x = = = =
D -2 - 1 3
1 1
1 - 2
2
1 k
D 1 - k -k - k 2k
y = = = =
D -2 - 1 3
1 1
1 - 2
k 2k
x = , y = , z = k , where k R
3 3
These values of x, y and z = k satisfy (iii) equation.