The document defines and provides examples of different types of matrices, including:
- Row, column, square, diagonal, scalar, unit, triangular, and transpose matrices. It also discusses properties of transpose matrices.
- Conjugate, tranjugate, symmetric, skew-symmetric, Hermitian, and skew-Hermitian matrices.
- Idempotent, nilpotent, and involutory matrices.
The document provides definitions, examples, and key properties for understanding different types of matrices used in linear algebra.
This document provides an overview of matrices including:
- How to describe matrices using m rows and n columns
- Common types of matrices such as row, column, zero, square, diagonal, and unit matrices
- Basic matrix operations including addition, subtraction, scalar multiplication
- Rules for matrix multiplication including that matrices must be conformable
- The transpose of a matrix which is obtained by interchanging rows and columns
- Properties of transposed matrices including (A+B)T = AT + BT and (AB)T = BTAT
This document provides an overview of matrices and quantitative techniques in management. It covers key topics related to matrices including:
- Defining matrices and examples of different types of matrices such as square, non-square, and vector matrices.
- Arithmetic operations that can be performed on matrices including addition, subtraction, and multiplication.
- Determinants and how to calculate the determinant of 2x2 and 3x3 matrices.
- The inverse of matrices and how to calculate the inverse of 2x2 and 3x3 matrices.
- How matrices can be used to represent and solve business problems involving quantities such as production levels.
1) A singular matrix has a determinant of 0, while a non-singular matrix has a non-zero determinant.
2) A symmetric matrix is equal to its transpose, while a skew-symmetric matrix is equal to the negative of its transpose.
3) The adjoint of a matrix is obtained by swapping diagonal entries and changing the sign of non-diagonal entries. For 3x3 matrices, the adjoint is the transpose of the cofactors.
It contains the basics of matrix which includes matrix definition,types of matrices,operations on matrices,transpose of matrix,symmetric and skew symmetric matrix,invertible matrix,
application of matrix.
The document discusses determinants and their properties. It defines determinants as representing single numbers obtained by multiplying and adding matrix elements in a special way. It then provides formulas for calculating determinants of matrices of order 1, 2 and 3. It also outlines several properties of determinants, such as how interchanging rows/columns, multiplying rows by constants, and adding rows affects the determinant. Finally, it discusses how determinants are used to determine whether systems of linear equations are consistent or inconsistent.
The document discusses matrices and their operations. It defines what a matrix is, provides examples of different types of matrices, and covers key matrix operations like addition, subtraction, scalar multiplication, and matrix multiplication. It also defines important matrix concepts such as the transpose of a matrix, inverse of a matrix, and properties related to these operations and concepts.
This document provides an overview of matrices including:
- How to describe matrices using m rows and n columns
- Common types of matrices such as row, column, zero, square, diagonal, and unit matrices
- Basic matrix operations including addition, subtraction, scalar multiplication
- Rules for matrix multiplication including that matrices must be conformable
- The transpose of a matrix which is obtained by interchanging rows and columns
- Properties of transposed matrices including (A+B)T = AT + BT and (AB)T = BTAT
This document provides an overview of matrices and quantitative techniques in management. It covers key topics related to matrices including:
- Defining matrices and examples of different types of matrices such as square, non-square, and vector matrices.
- Arithmetic operations that can be performed on matrices including addition, subtraction, and multiplication.
- Determinants and how to calculate the determinant of 2x2 and 3x3 matrices.
- The inverse of matrices and how to calculate the inverse of 2x2 and 3x3 matrices.
- How matrices can be used to represent and solve business problems involving quantities such as production levels.
1) A singular matrix has a determinant of 0, while a non-singular matrix has a non-zero determinant.
2) A symmetric matrix is equal to its transpose, while a skew-symmetric matrix is equal to the negative of its transpose.
3) The adjoint of a matrix is obtained by swapping diagonal entries and changing the sign of non-diagonal entries. For 3x3 matrices, the adjoint is the transpose of the cofactors.
It contains the basics of matrix which includes matrix definition,types of matrices,operations on matrices,transpose of matrix,symmetric and skew symmetric matrix,invertible matrix,
application of matrix.
The document discusses determinants and their properties. It defines determinants as representing single numbers obtained by multiplying and adding matrix elements in a special way. It then provides formulas for calculating determinants of matrices of order 1, 2 and 3. It also outlines several properties of determinants, such as how interchanging rows/columns, multiplying rows by constants, and adding rows affects the determinant. Finally, it discusses how determinants are used to determine whether systems of linear equations are consistent or inconsistent.
The document discusses matrices and their operations. It defines what a matrix is, provides examples of different types of matrices, and covers key matrix operations like addition, subtraction, scalar multiplication, and matrix multiplication. It also defines important matrix concepts such as the transpose of a matrix, inverse of a matrix, and properties related to these operations and concepts.
- 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.
A matrix is a rectangular array of numbers arranged in rows and columns. The order of a matrix describes its dimensions as the number of rows and columns. Matrices can be added or multiplied if they are the same order. Matrix multiplication results in a matrix whose order is the number of rows of the first matrix and columns of the second. The inverse of a square matrix A exists when AA-1=I=A-1A, where I is the identity matrix. The inverse is used to solve systems of linear equations of the form Ax=b by computing x=(A-1)b. Excel can perform matrix operations like multiplication and inversion using formulas.
The document describes the cofactor method for finding the inverse of a matrix A. It defines the cofactor Cij as the signed determinant of the matrix made by removing row i and column j from A. The inverse is then given by the transpose of the matrix of cofactors divided by the determinant of A. An example calculates the inverse of the 2x2 matrix A = [a, c; b, d] using this method.
This document provides an overview of matrix algebra concepts including:
- Matrix addition is defined as adding corresponding elements and is commutative and associative.
- Matrix multiplication is defined as taking the dot product of rows and columns. It is associative but not commutative.
- The transpose of a matrix is obtained by flipping rows and columns.
- Properties of matrix operations like addition, multiplication, and transposition are discussed.
Please go through the slides. It is very interesting way to learn this chapter for 2020-21.If you like this PPT please put a thanks message in my number 9826371828.
The document defines matrices and provides examples of different types of matrices. It discusses key concepts such as rows, columns, dimensions, entries, addition, subtraction, and multiplication of matrices. It also covers special matrices like identity matrices, inverse matrices, transpose of matrices, and using matrices to solve systems of linear equations. The document is a comprehensive overview of matrices that defines fundamental terms and concepts.
The document presents information on matrices, including:
- Definitions of matrices as rectangular arrangements of numbers arranged in rows and columns
- Common matrix operations such as addition, subtraction, scalar multiplication, and matrix multiplication
- Determinants and inverses of matrices
- How matrices can represent systems of linear equations
- Unique properties of matrices, such as the product of two non-zero matrices possibly being zero
- Applications of matrices in fields like geology, statistics, economics, and animation
This presentation describes Matrices and Determinants in detail including all the relevant definitions with examples, various concepts and the practice problems.
Matrices can be added, subtracted, and multiplied under certain conditions.
Addition and subtraction require matrices to be the same size.
Matrix multiplication requires the number of columns of the first matrix to equal the number of rows of the second matrix.
Matrices can also be multiplied by scalars.
This document discusses two methods for solving systems of linear equations using matrix algebra: Cramer's Rule and the inverse matrix method.
Cramer's Rule is described as a simple method that uses determinants and the inverse of the coefficient matrix to solve systems of linear equations. An example is shown working through applying Cramer's Rule to solve a 3x3 system.
The inverse matrix method is also explained. It involves converting the system of equations to matrix form Ax=b, then solving the equivalent inverse form x=A^-1b to find the solution values. An example 2x2 system is worked through using the inverse matrix method.
In summary, the document outlines Cramer's Rule and the inverse
This document defines and describes different types of matrices including:
- Upper and lower triangular matrices
- Determinants which are scalars obtained from products of matrix elements according to constraints
- Band matrices which are sparse matrices with nonzero elements confined to diagonals
- Transpose matrices which exchange the rows and columns of a matrix
- Inverse matrices which when multiplied by the original matrix produce the identity matrix
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.
The document defines key concepts in set theory including:
1. A set is a well-defined collection of distinct objects called elements. Georg Cantor is credited with creating set theory.
2. Sets can be represented in roster form by listing elements within curly brackets or in set-builder form using properties the elements share.
3. Operations on sets include intersection, union, difference, and complement. Intersection is the set of common elements, union is all elements in either set, difference is elements only in the first set, and complement is elements not in the set.
The document discusses various types of matrices:
- Row and column matrices are matrices with only one row or column respectively.
- A square matrix has the same number of rows and columns.
- A diagonal matrix has non-zero elements only along its main diagonal.
- An identity matrix has ones along its main diagonal and zeros elsewhere.
- A scalar matrix has all elements along its main diagonal multiplied by a scalar.
- A null matrix has all elements equal to zero.
The document also discusses properties such as the transpose of a matrix, symmetric matrices, and how to add, subtract and multiply matrices.
The document discusses matrices and their properties. It defines minors as the determinants of submatrices formed by deleting a row and column from the original matrix. Cofactors are defined as signed minors, with the sign determined by the parity of the sum of the row and column indices. The adjoint and inverse of matrices are discussed, with the inverse existing only for nonsingular matrices where the determinant is non-zero. Properties of the inverse include that the inverse of the inverse is the original matrix and that the inverse of a product is the product of the inverses in reverse order. Examples are provided to demonstrate calculating the adjoint, inverse and checking the inverse property.
The document defines various types of matrices including row vectors, column vectors, submatrices, square matrices, triangular matrices, diagonal matrices, identity matrices, zero matrices, and diagonally dominant matrices. It provides examples of each type of matrix. It also discusses when two matrices are considered equal, which is when they have the same size and corresponding elements are equal.
Here are the key steps to find the eigenvalues of the given matrix:
1) Write the characteristic equation: det(A - λI) = 0
2) Expand the determinant: (1-λ)(-2-λ) - 4 = 0
3) Simplify and factor: λ(λ + 1)(λ + 2) = 0
4) Find the roots: λ1 = 0, λ2 = -1, λ3 = -2
Therefore, the eigenvalues of the given matrix are -1 and -2.
The document discusses submatrices of circuit matrices and their properties. Some key points:
- A subgraph's incidence matrix is a submatrix of the original graph's incidence matrix.
- An (n-1)x(n-1) submatrix of a connected graph's incidence matrix is nonsingular if and only if the corresponding edges form a spanning tree.
- A circuit matrix represents cycles in a graph, with 1s indicating which edges are in each cycle.
- The circuit and incidence matrices of a graph have orthogonal rows, meaning their row dot products are all 0.
- A fundamental circuit matrix contains a basis of fundamental cycles, and has an identity submatrix, indicating its rank equals
This document discusses various claims and truths about digital advertising. It presents several "lies" about metrics like audience reach and advertising costs that are then countered by linked "truths" that provide more accurate or nuanced information. The overall message seems to be that while new digital advertising platforms provide opportunities to reach large audiences at low costs, advertisers need to provide valuable information and target their messages appropriately to make the most of these technologies.
Matrix is an Israeli IT company and the leading provider of technology services and solutions in Israel. It has 4,200 employees, $347M in annual revenue, and international presence. Matrix focuses on sectors like financial, telecom, government, and healthcare through specialized services, software products, and 19 Centers of Excellence staffed by experts in critical technologies. The Centers of Excellence help customers implement solutions more efficiently using best practices and expertise.
- 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.
A matrix is a rectangular array of numbers arranged in rows and columns. The order of a matrix describes its dimensions as the number of rows and columns. Matrices can be added or multiplied if they are the same order. Matrix multiplication results in a matrix whose order is the number of rows of the first matrix and columns of the second. The inverse of a square matrix A exists when AA-1=I=A-1A, where I is the identity matrix. The inverse is used to solve systems of linear equations of the form Ax=b by computing x=(A-1)b. Excel can perform matrix operations like multiplication and inversion using formulas.
The document describes the cofactor method for finding the inverse of a matrix A. It defines the cofactor Cij as the signed determinant of the matrix made by removing row i and column j from A. The inverse is then given by the transpose of the matrix of cofactors divided by the determinant of A. An example calculates the inverse of the 2x2 matrix A = [a, c; b, d] using this method.
This document provides an overview of matrix algebra concepts including:
- Matrix addition is defined as adding corresponding elements and is commutative and associative.
- Matrix multiplication is defined as taking the dot product of rows and columns. It is associative but not commutative.
- The transpose of a matrix is obtained by flipping rows and columns.
- Properties of matrix operations like addition, multiplication, and transposition are discussed.
Please go through the slides. It is very interesting way to learn this chapter for 2020-21.If you like this PPT please put a thanks message in my number 9826371828.
The document defines matrices and provides examples of different types of matrices. It discusses key concepts such as rows, columns, dimensions, entries, addition, subtraction, and multiplication of matrices. It also covers special matrices like identity matrices, inverse matrices, transpose of matrices, and using matrices to solve systems of linear equations. The document is a comprehensive overview of matrices that defines fundamental terms and concepts.
The document presents information on matrices, including:
- Definitions of matrices as rectangular arrangements of numbers arranged in rows and columns
- Common matrix operations such as addition, subtraction, scalar multiplication, and matrix multiplication
- Determinants and inverses of matrices
- How matrices can represent systems of linear equations
- Unique properties of matrices, such as the product of two non-zero matrices possibly being zero
- Applications of matrices in fields like geology, statistics, economics, and animation
This presentation describes Matrices and Determinants in detail including all the relevant definitions with examples, various concepts and the practice problems.
Matrices can be added, subtracted, and multiplied under certain conditions.
Addition and subtraction require matrices to be the same size.
Matrix multiplication requires the number of columns of the first matrix to equal the number of rows of the second matrix.
Matrices can also be multiplied by scalars.
This document discusses two methods for solving systems of linear equations using matrix algebra: Cramer's Rule and the inverse matrix method.
Cramer's Rule is described as a simple method that uses determinants and the inverse of the coefficient matrix to solve systems of linear equations. An example is shown working through applying Cramer's Rule to solve a 3x3 system.
The inverse matrix method is also explained. It involves converting the system of equations to matrix form Ax=b, then solving the equivalent inverse form x=A^-1b to find the solution values. An example 2x2 system is worked through using the inverse matrix method.
In summary, the document outlines Cramer's Rule and the inverse
This document defines and describes different types of matrices including:
- Upper and lower triangular matrices
- Determinants which are scalars obtained from products of matrix elements according to constraints
- Band matrices which are sparse matrices with nonzero elements confined to diagonals
- Transpose matrices which exchange the rows and columns of a matrix
- Inverse matrices which when multiplied by the original matrix produce the identity matrix
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.
The document defines key concepts in set theory including:
1. A set is a well-defined collection of distinct objects called elements. Georg Cantor is credited with creating set theory.
2. Sets can be represented in roster form by listing elements within curly brackets or in set-builder form using properties the elements share.
3. Operations on sets include intersection, union, difference, and complement. Intersection is the set of common elements, union is all elements in either set, difference is elements only in the first set, and complement is elements not in the set.
The document discusses various types of matrices:
- Row and column matrices are matrices with only one row or column respectively.
- A square matrix has the same number of rows and columns.
- A diagonal matrix has non-zero elements only along its main diagonal.
- An identity matrix has ones along its main diagonal and zeros elsewhere.
- A scalar matrix has all elements along its main diagonal multiplied by a scalar.
- A null matrix has all elements equal to zero.
The document also discusses properties such as the transpose of a matrix, symmetric matrices, and how to add, subtract and multiply matrices.
The document discusses matrices and their properties. It defines minors as the determinants of submatrices formed by deleting a row and column from the original matrix. Cofactors are defined as signed minors, with the sign determined by the parity of the sum of the row and column indices. The adjoint and inverse of matrices are discussed, with the inverse existing only for nonsingular matrices where the determinant is non-zero. Properties of the inverse include that the inverse of the inverse is the original matrix and that the inverse of a product is the product of the inverses in reverse order. Examples are provided to demonstrate calculating the adjoint, inverse and checking the inverse property.
The document defines various types of matrices including row vectors, column vectors, submatrices, square matrices, triangular matrices, diagonal matrices, identity matrices, zero matrices, and diagonally dominant matrices. It provides examples of each type of matrix. It also discusses when two matrices are considered equal, which is when they have the same size and corresponding elements are equal.
Here are the key steps to find the eigenvalues of the given matrix:
1) Write the characteristic equation: det(A - λI) = 0
2) Expand the determinant: (1-λ)(-2-λ) - 4 = 0
3) Simplify and factor: λ(λ + 1)(λ + 2) = 0
4) Find the roots: λ1 = 0, λ2 = -1, λ3 = -2
Therefore, the eigenvalues of the given matrix are -1 and -2.
The document discusses submatrices of circuit matrices and their properties. Some key points:
- A subgraph's incidence matrix is a submatrix of the original graph's incidence matrix.
- An (n-1)x(n-1) submatrix of a connected graph's incidence matrix is nonsingular if and only if the corresponding edges form a spanning tree.
- A circuit matrix represents cycles in a graph, with 1s indicating which edges are in each cycle.
- The circuit and incidence matrices of a graph have orthogonal rows, meaning their row dot products are all 0.
- A fundamental circuit matrix contains a basis of fundamental cycles, and has an identity submatrix, indicating its rank equals
This document discusses various claims and truths about digital advertising. It presents several "lies" about metrics like audience reach and advertising costs that are then countered by linked "truths" that provide more accurate or nuanced information. The overall message seems to be that while new digital advertising platforms provide opportunities to reach large audiences at low costs, advertisers need to provide valuable information and target their messages appropriately to make the most of these technologies.
Matrix is an Israeli IT company and the leading provider of technology services and solutions in Israel. It has 4,200 employees, $347M in annual revenue, and international presence. Matrix focuses on sectors like financial, telecom, government, and healthcare through specialized services, software products, and 19 Centers of Excellence staffed by experts in critical technologies. The Centers of Excellence help customers implement solutions more efficiently using best practices and expertise.
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The document discusses arithmetic and geometric sequences. It defines a sequence as a set of numbers in order. Sequences can be finite or infinite. Arithmetic sequences have a common difference between terms, while geometric sequences multiply each term by a constant ratio. Formulas are provided to calculate specific terms and the sum of terms in arithmetic and geometric sequences.
Lightning Talk #9: How UX and Data Storytelling Can Shape Policy by Mika Aldabaux singapore
How can we take UX and Data Storytelling out of the tech context and use them to change the way government behaves?
Showcasing the truth is the highest goal of data storytelling. Because the design of a chart can affect the interpretation of data in a major way, one must wield visual tools with care and deliberation. Using quantitative facts to evoke an emotional response is best achieved with the combination of UX and data storytelling.
This document summarizes a study of CEO succession events among the largest 100 U.S. corporations between 2005-2015. The study analyzed executives who were passed over for the CEO role ("succession losers") and their subsequent careers. It found that 74% of passed over executives left their companies, with 30% eventually becoming CEOs elsewhere. However, companies led by succession losers saw average stock price declines of 13% over 3 years, compared to gains for companies whose CEO selections remained unchanged. The findings suggest that boards generally identify the most qualified CEO candidates, though differences between internal and external hires complicate comparisons.
This document discusses matrices and their applications in solving systems of linear equations. It begins with defining what a matrix is and some special types of matrices including square, identity, zero, diagonal, commutative, triangular, and symmetric matrices. It also covers matrix operations like transpose and properties of determinants. The document then discusses using matrices to solve systems of linear equations through homogeneous systems, Gaussian elimination, Gaussian-Jordan elimination, and Cramer's rule. It includes examples and equations to illustrate the concepts and applications of matrices.
Matrices and its Applications to Solve Some Methods of Systems of Linear Equa...Rwan Kamal
This document is a research project submitted to the University of Duhok examining matrices and their applications in solving systems of linear equations. The project begins with an introduction to basic matrix concepts such as types of matrices, operations on matrices, and finding the inverse and determinant of matrices. It then discusses how matrices can be used to solve different types of systems of linear equations, including homogeneous systems, Gaussian elimination, Gaussian-Jordan elimination, and Cramer's rule. The document contains acknowledgments, contents, abstract, and introduction sections before presenting the technical matrix concepts and methods for solving systems of linear equations.
Matrices and its Applications to Solve Some Methods of Systems of Linear Equa...Abdullaا Hajy
This document is a research project submitted to the University of Duhok examining matrices and their applications in solving systems of linear equations. The project begins with an introduction to basic matrix concepts such as types of matrices, operations on matrices, and finding the inverse and determinant of matrices. It then discusses how matrices can be used to solve different types of systems of linear equations, including homogeneous systems, Gaussian elimination, Gaussian-Jordan elimination, and Cramer's rule. The document contains acknowledgments, contents, abstract, and introduction sections before presenting the technical matrix concepts and methods for solving systems of linear equations.
Some types of matrices, Eigen value , Eigen vector, Cayley- Hamilton Theorem & applications, Properties of Eigen values, Orthogonal matrix , Pairwise orthogonal, orthogonal transformation of symmetric matrix, denationalization of a matrix by orthogonal transformation (or) orthogonal deduction, Quadratic form and Canonical form , conversion from Quadratic to Canonical form, Order, Index Signature, Nature of canonical form.
Matrices and their operations were discussed. Key points include:
1) A matrix is a rectangular array of numbers. The order of a m x n matrix refers to its m rows and n columns.
2) Common matrix types include row/column matrices (vectors), square matrices, diagonal matrices, scalar matrices, identity matrices, and zero matrices.
3) Basic matrix operations include addition, subtraction, multiplication by a scalar, transpose, and multiplication. Properties like commutativity, associativity, and distributivity apply.
Definitions matrices y determinantes fula 2010 english subirHernanFula
The document discusses the history and properties of matrices. It describes how matrices were first introduced in 1850 and how their use has expanded. It then defines key matrix terms and concepts such as order, elements, types of matrices (e.g. triangular, diagonal), operations (e.g. addition, multiplication, inverse), and properties (e.g. of symmetric, banded and transpose matrices). It also provides examples of calculating the determinant of matrices using Sarrus' rule.
The document discusses matrices and their properties. It begins by defining a matrix as an ordered rectangular array of numbers or functions. It then discusses the order of a matrix, types of matrices including column, row, square, diagonal and identity matrices. It also defines equality of matrices. The key points are that a matrix represents tabular data, its order is given by the number of rows and columns, and there are different types of matrices based on their properties.
This document provides an overview of matrices and matrix operations. It defines what a matrix is and discusses matrix order and elements. It then covers basic matrix operations like scalar multiplication, addition, and multiplication. It introduces the concepts of transpose, special matrices like diagonal and triangular matrices, and the null and identity matrices. The document aims to define fundamental matrix concepts and arithmetic operations.
This document provides definitions and examples of different types of matrices including: real matrix, square matrix, row matrix, column matrix, null matrix, sub-matrix, diagonal matrix, scalar matrix, unit matrix, upper triangular matrix, lower triangular matrix, triangular matrix, single element matrix, equal matrices, singular and non-singular matrices. It also discusses elementary row and column transformations, rank of a matrix, solutions to homogeneous and non-homogeneous systems of linear equations, characteristic equations, eigenvectors and eigenvalues.
Beginning direct3d gameprogrammingmath05_matrices_20160515_jintaeksJinTaek Seo
This document provides an overview of linear systems and matrices. It defines key linear algebra concepts such as linear functions, linear maps, homogeneous and non-homogeneous linear systems, and plane equations. It also explains how to represent linear systems using matrices and describes common matrix operations including addition, scalar multiplication, transposition, and matrix multiplication. Finally, it discusses inverses, determinants, and using matrices to represent transformations such as rotations in 2D space.
This document discusses properties of symmetric, skew-symmetric, and orthogonal matrices. It defines each type of matrix and provides examples. Key points include:
- Symmetric matrices have Aij = Aji for all i and j. Skew-symmetric matrices have Aij = -Aji. Orthogonal matrices satisfy AT = A-1.
- The eigenvalues of symmetric matrices are always real. The eigenvalues of skew-symmetric matrices are either zero or purely imaginary.
- Any real square matrix can be written as the sum of a symmetric matrix and skew-symmetric matrix.
This document outlines topics related to matrices, including:
- Types of matrices such as real, square, row, column, null, sub, diagonal, scalar, unit, upper triangular, lower triangular, and singular matrices
- Characteristic equations, eigenvectors, and eigenvalues of matrices
- Properties of eigenvalues including that the sum of eigenvalues is the trace and the product is the determinant
- Examples of finding the sum and product of eigenvalues without directly calculating them
The document provides definitions and examples of key matrix concepts.
The document provides an overview of matrix theory, including:
1. The definition and notation of matrices, including that a matrix A is represented as Am×n, where m is the number of rows and n is the number of columns.
2. The different types of matrices and operations that can be performed on matrices, such as scalar multiplication, matrix multiplication, and properties like the distributive law.
3. Methods for solving systems of linear equations using matrices, including writing the system in matrix form, reducing the augmented matrix to echelon form, and determining the solution based on the rank.
The document defines various types of matrices and matrix operations. It discusses the definitions of a matrix, types of matrices including rectangular, column, row, square, diagonal and unit matrices. It also defines transpose, symmetric, skew-symmetric and cofactor matrices. The document provides examples of calculating the minor and cofactor of matrix elements, as well as the adjoint and inverse of matrices using the formula for the inverse of a square matrix in terms of its determinant and adjoint.
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.
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.
This document provides an overview of matrix algebra concepts for business students. It defines key terms like matrix, order, types of matrices including identity, diagonal and triangular matrices, and matrix operations such as addition, subtraction and multiplication. It also explains determinants, which evaluate whether a system of linear equations has a unique solution. Determinants are calculated by taking the difference of products of diagonal elements of a square matrix. This document serves as a basic introduction and recap of matrix algebra.
1) A matrix is a rectangular array of numbers arranged in rows and columns. The dimensions of a matrix are specified by the number of rows and columns.
2) The inverse of a square matrix A exists if and only if the determinant of A is not equal to 0. The inverse of A, denoted A^-1, is the matrix that satisfies AA^-1 = A^-1A = I, where I is the identity matrix.
3) For two matrices A and B to be inverses, their product must result in the identity matrix regardless of order, i.e. AB = BA = I. This shows that one matrix undoes the effect of the other.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
The order of the given matrix is 2×3. So the maximum no. of elements is 2×3 = 6.
The correct option is B.
The element a32 belongs to 3rd row and 2nd column.
The correct option is B.
3. A matrix whose each diagonal element is unity and all other elements are zero is called
A) Identity matrix B) Unit matrix C) Scalar matrix D) Diagonal matrix
4. A matrix whose each row sums to unity is called
A) Row matrix B) Column matrix C) Unit matrix D) Stochastic matrix
5. The sum of all the elements on the principal diagonal of a square
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.
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.
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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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How to Make a Field Mandatory in Odoo 17Celine George
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
How to Fix the Import Error in the Odoo 17Celine George
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
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 ...
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty,
International FDP on Fundamentals of Research in Social Sciences
at Integral University, Lucknow, 06.06.2024
By Dr. Vinod Kumar Kanvaria
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
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.
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MATRICES
MATRICES
A system of mn numbers arranged in a rectangular array of m rows and n columns is called an m by n matrix,
which is written as m × n matrix.
⎡a 11 a 12 ...a 1j ...a 1n ⎤
⎢ ⎥
⎢a 21 a 22 ...a 2 j ...a 2n ⎥
⎢.... .... .... . ... ⎥
Thus A = ⎢ ⎥ is a matrix of order m x n. It has m rows and n columns.
⎢a i1 a i2 ...a ij ...a in ⎥
⎢ ⎥
⎢.... .... .... . ... ⎥
⎢a a m2 a mj ...a mn ⎥
⎣ m1 ⎦
Note: The element aij is the element in the ith row and jth column of A
TYPES OF MATRICES
Row Matrix: A matrix having a single row is called a row matrix e.g., [1 3 5 7].
⎡ 2⎤
⎢ ⎥
Column Matrix: A matrix having a singe column is called a column matrix, e.g. ⎢3⎥
⎢5⎥
⎣ ⎦
Square matrix: A matrix having n rows and n columns is called a square matrix of order n.
Diagonal matrix: A square matrix all of whose elements except those in the leading diagonal are zero is
called a diagonal matrix.
⎡3 0 0⎤
⎢ ⎥
Ex: ⎢0 − 2 0⎥
⎢0 0
⎣ 6⎥
⎦
Scalar matrix: A diagonal matrix whose all the leading diagonal elements are equal is called a scalar matrix.
⎡3 0 0⎤
⎢ ⎥
Ex: ⎢0 3 0⎥
⎢0 0 3⎥
⎣ ⎦
Unit matrix: A diagonal matrix of order n which has unity for all its diagonal elements, is called a unit matrix or
an identity matrix of order n and is denoted by In.
⎡1 0 0⎤
⎢ ⎥
For example, unit matrix of order 3 is ⎢0 1 0⎥ .
⎢0 0 1⎥
⎣ ⎦
Triangular matrix: A square matrix all of whose elements below the leading diagonal are zero is called an
upper triangular matrix.
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A square matrix all of whose elements above the leading diagonal are zero is called a lower triangular
matrix.
⎡a h g⎤ ⎡1 0 0⎤
⎢ ⎥ ⎢ ⎥
Thus ⎢0 b f ⎥ and ⎢2 3 0 ⎥ are upper and lower triangular matrices respectively
⎢0 0 c ⎥
⎣ ⎦ ⎢1 − 5
⎣ 4⎥⎦
Transpose of a matrix: The transpose of a given matrix A is obtained by interchanging the rows and columns
of the matrix A. Transpose of A is denoted by A' or A'.
Thus, if A = [aij]mxn Then A' = [bij]nxm Where bij = aji
⎛1 2⎞
If A = ⎛
1 3 5⎞
Ex. ⎜
⎜ ⎟
⎟
Then A' = ⎜ 3 7 ⎟
⎜ ⎟
⎝2 7 0⎠ ⎜5
⎝ 0⎟
⎠
Properties of Transpose: Let A and B be two matrices. Then
(i) (AT)T = A
(ii) (A + B)T = AT + BT, A and B being of the same order.
(iii) (kA)T = kAT, k be any scalar (real or complex).
(iv) (AB)T = BTAT, A and B being conformable for the product AB.
Conjugate of a Matrix: The conjugate of a given matrix A is obtained by replacing the elements of A by their
corresponding complex conjugates. The conjugate of A is denoted by A .
Thus, if A = [aij]mxn then A = [ a ij]mxn
1 1 +i⎞ ⎛1 1 − i ⎞
Ex: If A = ⎛
⎜
⎜ ⎟ then A = ⎜
⎟ ⎜ ⎟
⎟
⎝3 1 −1⎠ ⎝3 1 + 1⎠
Tranjugate or Transposed Conjugate of a Matrix: The transpose conjugate of a given matrix is obtained by
interchanging the rows and columns of the matrix obtained by replacing the element of A by their
corresponding complex conjugate. The transpose conjugate of A is denoted by A* or by A θ . Thus,
A* = ( A ' ) = ( A )’
⎛ 2 3⎞
2 1+ i 0⎞ ⎜ ⎟
Ex. If A = ⎛
⎜
⎜ ⎟ then A* = ⎜1− i 2 ⎟
⎟
⎝3 2 i⎠ ⎜ 0
⎝ − i⎟
⎠
Symmetric and skew-symmetric matrices: A square matrix A = [aij] is said to be symmetric when aij = aji for
all i and j.
If aij = – aji for all i and j so that all the leading diagonal elements are zero, then the matrix is called a skew-
symmetric matrix.
Examples of symmetric and skew-symmetric matrices are respectively.
⎡a h g⎤ ⎡ 0 h − g⎤
⎢ ⎥ ⎢ ⎥
⎢h b f ⎥ and ⎢− h 0 f ⎥
⎢
⎣g f c ⎥
⎦ ⎢ g −f 0 ⎥
⎣ ⎦
Note: Every square matrix can be uniquely expressed as a sum of a symmetric and a skew-symmetric matrix
as:
A = ½ (A + A’) + ½ (A – A’).
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Hermitian Matrix: A square matrix A is said to be a Hermitian matrix if the transpose of the conjugate matrix
is equal to the matrix itself i.e., A* = A ⇒ a ij = aji, where A = [aij]nxn; aij ∈ C.
⎛ 3 2 + i 5i ⎞
⎛ 2 3 + 2i ⎞ ⎜ ⎟
Ex : ⎜
⎜ 3 − 2i ⎟ ⎜2 − i 0 2⎟
⎝ 7 ⎟ ⎠ ⎜− 5i
⎝ 2 7⎟ ⎠
Skew Hermitian Matrix: A square matrix A is said to be skew-Hermitian, if
A* = – A ⇒ a ij = – aji
⎛ 4i 2−i 3 ⎞
⎛ 2i 5 + 4i ⎞ ⎜ ⎟
Ex. ⎜
⎜ − 5 − 4i ⎟ ⎜− 2 + i 0 4 ⎟ are skew –Hermitian matrices.
⎝ 0 ⎟ ⎠ ⎜ −3
⎝ − 4 − 3i ⎟
⎠
Idempotent Matrix: A square matrix A, such that A2 = A is called idempotent matrix
⎛ 2 − 2 − 4⎞
⎜ ⎟
Ex. ⎜ − 1 3 4 ⎟ is idempotent because
⎜ 1 − 2 − 3⎟
⎝ ⎠
⎛ 2 − 2 − 4 ⎞ ⎛ 2 − 2 − 4 ⎞ ⎛ 4 + 2 − 4 − 4 − 6 + 4 − 8 − 8 + 12 ⎞
2 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
A = A. A = ⎜ − 1 3 4 ⎟ ⎜−1 3 4 ⎟ = ⎜− 2 − 3 + 4 2 + 9 − 8 4 + 12 − 12 ⎟
⎜ 1 − 2 − 3⎟ ⎜ 1 − 2 − 3⎟ ⎜ 2 + 2 − 3 − 2 − 6 + 6 − 4 − 8 + 9 ⎟
⎝ ⎠ ⎝ ⎠ ⎝ ⎠
⎛ 2 − 2 − 4⎞ ⎛ 2 − 2 − 4⎞ ⎛ 4 + 2 − 4 − 4 − 6 + 4 − 8 − 8 + 12 ⎞
2 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
A = A. A = ⎜ − 1 3 4 ⎟ ⎜− 1 3 4 ⎟ = ⎜− 2 − 3 + 4 2 + 9 − 8 4 + 12 − 12 ⎟
⎜ 1 − 2 − 3⎟ ⎜ 1 − 2 − 3⎟ ⎜ 2+2−3 −2−6+6 −4−8+9 ⎟
⎝ ⎠ ⎝ ⎠ ⎝ ⎠
⎛ 2 − 2 − 4⎞
⎜ ⎟
⎜− 1 3 4 ⎟= A
⎜ 1 − 2 − 3⎟
⎝ ⎠
Nilpotent Matrix: If A is a square matrix such that Am = 0, where m is a positive integer, than A is called a
nilpotent matrix. If m is the least positive integer for which Am = 0, then A is called to be nilpotent matrix of
index m.
⎛ 1 2 3 ⎞
⎜ ⎟ 2
Ex. The matrix A = ⎜ 1 2 3 ⎟ is a nilpotent matrix of index 2, because A = A . A
⎜− 1 − 2 − 3⎟
⎝ ⎠
⎛ 1 2 3 ⎞ ⎛ 1 2 3 ⎞ ⎛ 1+ 2 − 3 2+4−6 3 + 6 − 9 ⎞ ⎛0 0 0⎞
⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎜ 1 2 3 ⎟ × ⎜ 1 2 3 ⎟ = ⎜ 1+ 2 − 3 2+4−6 3 + 6 − 9 ⎟ = ⎜0 0 0⎟ = 0
⎜− 1 − 2 − 3⎟ ⎜− 1 − 2 − 3⎟ ⎜− 1− 2 + 3 − 2 − 2 + 6 − 3 − 6 + 9⎟ ⎜0 0 0⎟
⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
Involutory Matrix: A square matrix A such that A2 = I is called involutory matrix.
The matrix A = ⎛
1 0⎞ 2
Ex. ⎜
⎜ ⎟ is involutory because A = A . A
⎝ 0 − 1⎟
⎠
⎛ 1.1 + 0.0 1.0 + 0(−1) ⎞
= ⎛
0⎞ ⎛1 0 ⎞
= ⎛
1 1 0⎞
⎜
⎜ ⎟ × ⎜
⎟ ⎜ 0 − 1⎟
⎟
= ⎜
⎜ 0.1 + (−1).0 0.0 + (−1).(−1) ⎟
⎟ ⎜
⎜ ⎟
⎟
⎝ 0 − 1⎠ ⎝ ⎠ ⎝ ⎠ ⎝ 0 1⎠
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Orthogonal Matrix: A square matrix A is said to be orthogonal matrix, if AA' = I = A' A.
⎛−1 2 2⎞
1 ⎜ ⎟
Ex. The matrix ⎜ 2 −1 2 ⎟ is orthogonal matrix, because
3 ⎜ 2
⎝ 2 − 1⎟
⎠
⎛− 1 2 2⎞ ⎛− 1 2 2 ⎞ ⎛9 0 0⎞ ⎛1 0 0⎞
A A’ = 1 ⎜ 2 ⎟ 1 ⎜ ⎟ 1 ⎜ ⎟ ⎜ ⎟
−1 2 ⎟ x ⎜ 2 −1 2 ⎟ = ⎜0 9 0⎟ = ⎜0 1 0 ⎟ = I3.
3 ⎜ 3 ⎜ 9
⎜ 2
⎝ 2 − 1⎟
⎠ ⎝ 2 2 − 1⎟
⎠
⎜0
⎝ 0 9⎟
⎠
⎜0
⎝ 0 1⎟⎠
Similarly A'A = 1
Unitary Matrix: A square matrix A is said to be unitary matrix, if AA* = I = A*A.
1 ⎡ 1 1+i⎤ 1 ⎡ 1 1+i⎤ 1 ⎡ 1 1+i⎤
Ex. A= ⎢ ⎥ is an unitary matrix, because AA* = ⎢1−i −1⎥ x = ⎢ ⎥
3 ⎣1−i −1⎦ 3⎣ ⎦ 3 ⎣1−i −1⎦
1 ⎛1 + 1 − i 2 0 ⎞ 1 ⎛3 0⎞ ⎛ 1 0⎞
= ⎜ ⎟= ⎜ ⎟ = ⎜
⎜ 0 1⎟
= I. Similarly, A*A = I.
3 ⎜ 0
⎝ 1 − i 2 + 1⎟ 3 ⎜ 0 3 ⎟
⎠ ⎝ ⎠ ⎝
⎟
⎠
Singular Matrix: A square matrix A is called singular if A = 0.
Non singular Matrix: A square matrix A is called non-singular if A ≠ 0.
Properties of Matrices:
Addition and subtraction of matrices: If A, B be two matrices of the same order, then their sum A + B is
defined as the matrix each element of which is the sum of the corresponding elements of A and B
⎡a 1 b1 ⎤ ⎡c 1 d1 ⎤ ⎡a 1 + c 1 b 1 + d1 ⎤
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
Thus ⎢a 2 b 2 ⎥ + ⎢c 2 d 2 ⎥ = ⎢a 2 + c 2 b 2 + d 2 ⎥
⎢a 3
⎣ b3 ⎥⎦ ⎢c 3 d 3 ⎥
⎣ ⎦ ⎢a 3 + c 3 b 3 + d 3 ⎥
⎣ ⎦
Similarly A – B is defined as a matrix whose elements are obtained by subtracting the elements of B from the
corresponding elements of A.
⎡a 1 b1 ⎤ ⎡c 1 d1 ⎤ ⎡a 1 − c 1 b 1 − d1 ⎤
Thus ⎢ ⎥ – ⎢ ⎥ = ⎢ ⎥
⎣a 2 b2 ⎦ ⎣c 2 d 2 ⎦ ⎣a 2 − c 2 b 2 − d 2 ⎦
Equal matrices: Two matrices A and B are said to be equal, written as A = B, if
• They are both of the same order i.e. have the same number of rows and columns, and
• The elements in the corresponding places of the two matrices are the same
1. Only matrices of the same order can be added or subtracted.
2. Addition of matrices is commutative,
i.e. A + B = B + A.
3. Addition and subtraction of matrices is associative.
i.e. (A + B) – C = A + (B – C) = B + (A – C).
4. A – B = A + (– B)
5. A+B=A+C⇒B=C
6. A+0=A=0+A
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Multiplication of matrix by a scalar: The product of a matrix A by a scalar k is a matrix whose each element
is k times the corresponding elements of A.
⎡a b c ⎤ ⎡ka kb 1 kc 1 ⎤
Thus k ⎢ 1 1 1 ⎥ = ⎢ 1 ⎥
⎣a2 b2 c 2 ⎦ ⎣ka 2 kb 2 kc 2 ⎦
The distributive law holds for such products, i.e. k (A + B) = kA + kB.
All the laws of ordinary algebra hold for the addition or subtraction of matrices and their multiplication by
scalars. Therefore
• (k + l ) A = kA + l A
• (k l ) A = k ( l A) = l (kA)
• l A = A l , l ∈ R (or C)
• k (A + B) = kA + kB
Ex: Evaluate 3A – 4B, where
⎡3 −4 6⎤ ⎡1 0 1 ⎤
A= ⎢ ⎥ and B = ⎢ ⎥
⎣5 1 7⎦ ⎣2 0 3 ⎦
⎡9 − 12 18⎤ ⎡4 0 4⎤
We have 3A = ⎢ ⎥ and 4B = ⎢ ⎥
⎣15 3 21 ⎦ ⎣8 0 12 ⎦
⎡9 − 4 − 12 − 0 18 − 4 ⎤ ⎡5 − 12 14⎤
∴ 3A – 4B = ⎢ ⎥ = ⎢ ⎥
⎣15 − 8 3−0 21 − 12⎦ ⎣7 3 9⎦
Multiplication of matrices: Two matrices can be multiplied only when the number of columns in the first is
equal to the number of rows in the second. Such matrices are said to be conformable.
⎛ 2 1 3⎞ ⎛1 − 2⎞
⎜ ⎟ ⎜ ⎟
For example, If A = ⎜ 3 −2 1 ⎟ and B = ⎜ 2 1 ⎟ , then A and B are conformable for the product AB such
⎜− 1 0 1⎟ ⎜4 − 3⎟
⎝ ⎠ ⎝ ⎠
⎛ 1⎞
that (AB)11 = (First row of A) (First column of B) = [2 1 3] ⎜ 2 ⎟ = 2*1 + 1*2 + 3*4 = 16 (AB) 12 = (First row of A)
⎜ ⎟
⎜ 4⎟
⎝ ⎠
⎛ − 2⎞ ⎛16 − 12⎞
(Second column of B) = [2 1 3] ⎜ ⎟ = 2* – 2 + 1*1 + 3 * – 3 = – 12 etc. Thus AB = ⎜ 3
⎜
⎟
− 11⎟ .
⎜ 1⎟
⎜ − 3⎟ ⎜3 −1 ⎟
⎝ ⎠ ⎝ ⎠
Properties of Matrix Multiplication:
• Matrix multiplication is associative i.e. if A, B, C are m x n, n x p and q x q matrices respectively, then (AB)
C = A (BC)
• Matrix multiplication is not always commutative AB ≠ BA
• Matrix multiplication is distributive over matrix addition i.e. A (B + C) = AB + AC where A, B, C are m x n, n
x p, n x p matrices respectively
• The product of two matrices can be the null matrix while neither of them is the null matrix.
For example, if A = ⎛
0 2⎞ ⎛ 1 0⎞ ⎛0 0⎞
⎜
⎜ ⎟ and B = ⎜
⎟ ⎜ ⎟ then AB = ⎜
⎟ ⎜ ⎟ while neither A nor B is a null matrix.
⎟
⎝0 0⎠ ⎝0 0⎠ ⎝0 0⎠
• Two matrix are said to be commute if AB = BA.
If AB = – BA, they are called Anti-commute.
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Power of a matrix: If A be a square matrix, then the product AA is defined as A2. Similarly, we define higher
powers of A. i.e. A.A2 = A3, A2. A2 = A4 etc.
⎡1 3 2⎤
⎢ ⎥
Exam. Prove that A3 – 4A2 – 3A + 11 I = 0, where A = ⎢2 0 − 1⎥ .
⎢1
⎣ 2 3⎥
⎦
⎡1 3 2⎤ ⎡1 3 2⎤ ⎡1 + 6 + 2 3 + 0 + 4 2 − 3 + 6⎤ ⎡9 7 5 ⎤
⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥
A = A × A = ⎢2
2
0 − 1⎥ × ⎢2 0 − 1⎥ = ⎢2 + 0 − 1 6 + 0 − 2 4 + 0 − 3 ⎥ = ⎢1 4 1 ⎥
⎢1
⎣ 2 3⎥
⎦ ⎢1
⎣ 2 3⎥
⎦ ⎢1 + 4 + 3 3 + 0 + 6 2 − 2 + 9 ⎥
⎣ ⎦ ⎢8 9 9⎥
⎣ ⎦
⎡9 7 5 ⎤ ⎡1 3 2⎤ ⎡9 + 14 + 5 27 + 0 + 10 18 − 7 + 15 ⎤ ⎡28 37 26⎤
⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥
A = A × A = ⎢1 4 1 ⎥ × ⎢2
3 2
0 − 1⎥ = ⎢1 + 8 + 1 3+0+2 2 − 4 + 3 ⎥ = ⎢10 5 1⎥
⎢8 9 9⎥
⎣ ⎦ ⎢1
⎣ 2 3⎥
⎦ ⎢8 + 18 + 9 24 + 0 + 18 16 − 9 + 27⎥
⎣ ⎦ ⎢35
⎣ 42 34 ⎥
⎦
A3 – 4A2 – 3A + 11 I
⎡28 37 26⎤ ⎡9 7 5 ⎤ ⎡1 3 2⎤ ⎡1 0 0⎤
⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢10 5 1⎥ – 4 ⎢1 4 1 ⎥ – 3 ⎢2 0 − 1⎥ + 11 ⎢0 1 0⎥
⎢35
⎣ 42 34 ⎥
⎦ ⎢8 9 9⎥
⎣ ⎦ ⎢1
⎣ 2 3⎥
⎦ ⎢0 0 1⎥
⎣ ⎦
⎡28 − 36 − 3 + 11, 37 − 28 − 9 + 0, 26 − 20 − 6 + 0 ⎤ ⎡0 0 0⎤
⎢ ⎥ ⎢ ⎥
= ⎢10 − 4 − 6 − 0, 5 − 16 + 0 + 11, 1 − 4 + 3 + 0 ⎥ = ⎢0 0 0⎥ = 0
⎢35 − 32 − 3 + 0, 42 − 36 − 6 + 0,
⎣ 34 − 36 − 9 + 11⎥
⎦ ⎢0 0 0⎥
⎣ ⎦
Adjoint of A Matrix: The determinant of the square matrix
a1 b1 c 1 a1 b1 c 1
A = a2 b2 c 2 is ∆ a 2 b 2 c2
a3 b3 c3 a3 b3 c3
A 1 B 1 C1
The matrix formed by the cofactors of the elements in ∆ is A 2 B 2 C2 .
A 3 B3 C3
A1 A 2 A3
Then, the transpose of this matrix, i.e. B 1 B 2 B3
C1 C 2 C3
is called the adjoint of the matrix A and is written as Adj. A.
Thus the Adjoint of A is the transposed matrix of cofactors of A.
Inverse of a matrix:
Let A be a square matrix of order n. If there exists a matrix B, such that
A. B = B. A = I, then B is called the inverse of the matrix and denoted by A–1.
The Inverse of a Square Matrix A exists if and only if A is non-singular (i.e., det A ≠ 0).
The Inverse a Square Matrix is unique.
−1 adj A
If A is a non-singular matrix then A =
det A
⎡a b⎤ −1 1 ⎡a b⎤
Example: If A = ⎢ ⎥ is non-singular then A = ⎢ ⎥
⎣c d⎦ ad − bc ⎣c d⎦
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⎡ 1 2⎤
A 33 = ( −1)3 + 3 ⎢ ⎥ = −5,
⎣3 1⎦
⎡− 2 − 2 2 ⎤
⎢ ⎥
∴ Matrix of co − factors = ⎢ 1 − 3 1 ⎥
⎢ 3 11 − 5⎥
⎣ ⎦
⎡− 2 1 3 ⎤
⎢ ⎥
Adj A = ⎢− 2 − 3 11 ⎥
⎢2
⎣ 1 − 5⎥⎦
⎡− 2 1 3 ⎤
Adj A 1 ⎢ ⎥
A −1 = = ⎢− 2 − 3 11 ⎥
det A 4
⎢2
⎣ 1 − 5⎥⎦
Coshα Sinhα
Example: If A = , find A −1
Sinhα Coshα
Coshα Sinhα
Solution: det A =
Sinhα Coshα
= cos h2α – sinh2α = 1 ≠ 0. ∴ A–1 can be computed
RANK OF A MATRIX:
If we select any r rows and r columns from any matrix A, deleting all the other rows and columns, then the
determinant formed by these r × r elements is called the minor of A of order r. Clearly there will be a number of
different minors of the same order, got by deleting different rows and columns form the same matrix.
Def. A matrix is said to be of rank r when
(i) it has at least one non-zero minor of order r,
and (ii) every minor of order higher than r vanishes.
Briefly, the rank of a matrix is the largest order of any non-vanishing minor of the matrix.
If a matrix has a non-zero minor of order r, its rank is ≥ r.
If all minors of a matrix of order r + 1 are zero, its rank is ≤ r.
Note:
1. Since the rank of every non-zero matrix is ≥ 1, we agree to assign the rank, zero to every null matrix.
2. Every (r + 1) rowed minor of a matrix can be expressed as the sum of its r-rowed minors Therefore if all
the r-rowed minors of a matrix are equal to zero, then obviously all its (r + 1) rowed minors will also be
equal to zero.
Important: The following two simple results will help us very much in finding the rank of a matrix.
(i) The rank of a matrix is ≤ r , if all (r + 1)-rowed minors of the matrix vanish
(ii) The rank of a matrix is ≥ r, if there is at least one r-rowed minor of the matrix which is not equal to zero.
Examples:
⎛ 1 0 0⎞
⎜ ⎟
1. Let A = I3 = ⎜ 0 1 0 ⎟ be a unit matrix of order 3.
⎜ 0 0 1⎟
⎝ ⎠
We have A = 1 Therefore A is a non-singular matrix Hence rank A = 3.In particular, the rank of a unit
matrix of order n is n
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⎛0 0 0⎞
⎜ ⎟
2. Let A = ⎜ 0 0 0 ⎟ Since A is a null matrix, therefore rank A = 0
⎜0 0 0⎟
⎝ ⎠
⎛1 2 3⎞
⎜ ⎟
3. Let A = ⎜ 2 3 4 ⎟ We have A = 1 (6 – 8) – 2(4 – 4) + 3(4 – 3) = 2 + 1 = 3. Thus A is a non-singular
⎜ 1 2 2⎟
⎝ ⎠
Therefore rank A = 3.
⎛ 1 2 3⎞
⎜ ⎟
4. Let A = ⎜ 3 4 5 ⎟ . We have A = 1 (24 – 25) – 2 (18 – 20) + 3 (15 – 16) = 0.
⎜ 4 5 6⎟
⎝ ⎠
1 2
Therefore the rank of A is less than a 3. Now there is at least one minor of A or order 2, namely
3 4
which is not equal to zero Hence rank A = 2.
⎛3 1 2⎞
⎜ ⎟
5. Let A = ⎜ 6 2 4 ⎟ we have A = 0, since the first two columns are identical Also each 2 – rowed
⎜3 1 2⎟
⎝ ⎠
minor of A is equal to zero. But A is not a null matrix. Hence rank A = 1
⎛ 2 4 3 2⎞ 2 4
6. Let A = ⎜
⎜3 5 1 4 ⎟
⎟ Here we see that there is at least one minor of A of order 2 i.e., which is
⎝ ⎠ 3 5
not equal to zero Also there is no minor of A of order greater that 2 Hence rank of A = 2
Echelon form of matrix: A matrix A is said to be Echelon from if
(i) Every row of A which has all its entries 0 occurs below every row, which has a non-zero entry.
(ii) The first non-zero entry in each non-zero row is equal to 1.
(iii) The number of zeros before the first non-zero element in a row is less than the number of such zeros in
the next row.
Important result:
The rank of a matrix in Echelon from is equal to the number of non-zero rows of the matrix.
⎛0 1 2 3 ⎞
⎜ ⎟
Ex Find the rank of the matrix ⎜ 0 0 1 − 1⎟
⎜0 0 0 0 ⎟
⎝ ⎠
Sol. The matrix A has zero row. We see that it occurs below every non-zero row. Further the number of zero
before the first non-zero element in the first row is one. The number of zeros before the first non-zero
element in the second row it two. Thus the number of zeros before the first non-zero element in any row
is less than the number of such zeros in the next row. Thus the matrix A is in Echelon from rank A =
the number of non-zero rows of A = 2.
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Examples
1. Find the rank of each of the following matrices:
⎛ 1 2 3⎞
⎜ ⎟
(ii) ⎛
1 2 3 ⎞
(i) ⎜ 2 1 0 ⎟ ⎜
⎜ ⎟
⎟
⎜ 0 1 2⎟ ⎝2 4 5⎠
⎝ ⎠
⎛ 1 2 3⎞
Sol. (i) Let A = ⎜ 2 1 0 ⎟ We have A = 1(2 – 0) + 3(2 – 0), expanding along the first row
⎜ ⎟
⎜0 1 2⎟
⎝ ⎠
= 2 – 8 + 6 = 0.
1 2
But there is at least one minor of order 2 of the matrix A, namely which is not equal to zero.
2 1
Hence the rank A = 2.
⎛1 2 3 ⎞ 1 3
(ii) ⎜ 2 4 5 ⎟ Here there is at least one minor of order 2 of the matrix A, namely 2 5 which
Let A = ⎜ ⎟
⎝ ⎠
is not equal to 0. Also there is not minor of the matrix A of order greater than 2. Hence rank A =
2.
2. Show that the rank of a matrix every element of which is unity is 1.
Sol. Let A denote a matrix every element of which is unity. All the 2-rewed minors of A obviously vanish but
A is a non-zero matrix. Hence rank A = 1.
3. A is a non-zero column and B a non-zero row matrix, show that rank (AB) = 1.
⎛ a 11 ⎞
⎜ ⎟
⎜ a 21 ⎟
Sol. A = ⎜ a ⎟ and B = (b 11 b 12 b 13 ....... b in )
⎜ 31 ⎟
⎜ ...... ⎟
⎜ ⎟
⎝ a ml ⎠
be two non-zero column and row matrices respectively.
⎛ a 11b 11 a 11b 12 a 11b 13 ...... a 11b 1n ⎞
⎜ ⎟
We have AB = ⎜ a 21b 11 a 21b 12 a 21b 13 ...... a 21b 1n ⎟
⎜ ...... ...... ...... ...... ...... ⎟
⎜ ⎟
⎜a b a ml b 12 a ml b 13 ...... a ml b 1n ⎟
⎝ ml 11 ⎠
Since A and B are non-zero matrices therefore the matrix AB will also be non-zero The matrix AB will
have at least one non zero element obtained by multiplying corresponding non-zero elements of A and
B All the two-rowed minors of A obviously vanish. But A is a non-zero matrix. Hence rank AB = 1.
⎛0 1 0 0⎞
⎜ ⎟
4. ⎜0 0 1 0 ⎟ find the ranks of U and U2
⎜0 0 0 1⎟
⎜ ⎟
⎜0 0 0 0⎟
⎝ ⎠
Sol. The matrix U is the Echelon form. rank U = the number of non-zero rows of U = 3.
⎛0 1 0 0⎞ ⎛0 1 0 0⎞ ⎛0 0 1 0⎞
⎜ ⎟ ⎜ ⎟ ⎜ ⎟
Now U =
2 ⎜0 0 1 0⎟ x ⎜0 0 1 0⎟ = ⎜0 0 1 0⎟
⎜0 0 0 1⎟ ⎜0 0 0 1⎟ ⎜0 0 0 0⎟
⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎜0 ⎟
0 0 0⎠ ⎜0 0 0 0⎟ ⎜0 0 0 0⎟
⎝ ⎝ ⎠ ⎝ ⎠
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Elementary Operations: An elementary operation is said to be row (or column) operation if it is applied to
rows (or columns). There are three types of elementary operations:
Type 1: The interchanging of ith row (or columns) of the matrix, this operation is denoted by
Ri <--> Rj or (Ci <--> Cj) or by Rij (Cij)
Type 2: If ith row (or column) of the matrix is multiplied by scalar K(K ≠ 0), then this row (or column)
operation is denoted by Ri <--> kRj (Ci <--> Cj)
Type 3: If k times of the elements of the ith row (or column) are added to the corresponding elements of jth
row (or column), then this row (column) operation is denoted by RJ RJ or kRi (or Cj Cj + kCj
Examples:
⎛ 1 0⎞
1. If I = ⎜
⎜ ⎟ then E12 – E21 is equal to:
⎟
⎝0 1⎠
⎛01 0⎞ ⎛ − 1 − 1⎞
(a) ⎛ 0 − 1⎞
⎜ ⎟ (b) ⎜
⎜ ⎟
⎟ (c) ⎜
⎜ ⎟
⎟ (d) None of these
⎜ ⎟
⎝− 1 0 ⎠ ⎝ 0 0⎠ ⎝ − 1 − 1⎠
⎛ 1 0⎞ ⎛0 1⎞ ⎛0 1⎞
Sol. Given that = ⎜
⎜ ⎟ ⎜ 1 0⎟ R1 < ⇒ R12 E21 =
⎟ Therefore E12 = ⎜ ⎟ ⎜ 1 0⎟ , R1 <⇒ R2
⎜ ⎟
⎝0 1⎠ ⎝ ⎠ ⎝ ⎠
⎛0 1⎞ ⎛ 0 1⎞ ⎛0 0⎞
E12 – E21 = ⎜
⎜ ⎟ ⎜
⎟ ⎜ ⎟= ⎜
⎟ ⎜ ⎟ Therefore the correct answer is (b).
⎟
⎝ 1 0⎠ ⎝ − 1 0⎠ ⎝ 0 0⎠
2. If I = ⎛ 1 0 ⎞ , then 2E12 (2) + 3 E21 is equal to:
⎜ ⎟
⎜ ⎟
⎝0 1⎠
(a) ⎛ 2 7 ⎞
⎜ ⎟ (b) ⎛ 7
⎜
2⎞
⎟ (c) ⎛ 2 7 ⎞
⎜ ⎟ (d) ⎛ 4
⎜
4⎞
⎟
⎜ ⎟ ⎜2 3⎟ ⎜ ⎟ ⎜3 1⎟
⎝2 3⎠ ⎝ ⎠ ⎝3 2⎠ ⎝ ⎠
Sol. E12 = ⎛ 1 2 ⎞ R1 → R1 + 2R2 E21 = ⎛ 0
⎜ ⎟ ⎜
1⎞
⎟ , R1 ⇔ R2
⎜ ⎟ ⎜ 0⎟
⎝0 1⎠ ⎝1 ⎠
Therefore 2E12 (3) + 3E21 = 2 ⎛ 1 2 ⎞ + 3 ⎛ 0
⎜ ⎟ ⎜
1⎞
⎟ = ⎛2 7⎞ .
⎜ ⎟
⎜ ⎟ ⎜ 0⎟ ⎜ ⎟
⎝0 1⎠ ⎝1 ⎠ ⎝3 2⎠
Therefore the correct answer is (c).
Elementary matrices: An elementary matrix is that, which is obtained from a unit matrix, by subjecting it to
any of the elementary transformations.
Examples of elementary matrices obtained from
⎡1 0 0⎤ ⎡1 0 0⎤ ⎡1 0 0⎤ ⎡1 p 0 ⎤
⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥
I3 = ⎢0 1 0⎥ are R23 ⎢0 0 1 ⎥ = C23; kR2 = ⎢0 k 0 ⎥ ; R1 + pR2 = ⎢0 1 0⎥
⎢0 0 1 ⎥
⎣ ⎦ ⎢0 1 0 ⎥
⎣ ⎦ ⎢0 0 1 ⎥
⎣ ⎦ ⎢0 0 1⎥
⎣ ⎦
Equivalent matrix: Two matrices A and b are said to be equivalent if one can be obtained from the other by a
sequence of elementary transformations. Two equivalent matrices have the same order and the same rank.
The symbol ~ is used for equivalence.
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IMPORTANT FACTS
1. The matrix obtained by applying a row operation on a given matrix is same as the matrix obtained by
pre-multiplication of the given matrix by the corresponding elementary matrix.
For Example: Let A = ⎛ 3
⎜
4⎞
⎟ Let us operate R12, then A ~ ⎛ 8
⎜
9⎞
⎟ ….. (1)
⎜8 9⎟ ⎜3 4⎟
⎝ ⎠ ⎝ ⎠
Now let 1 = ⎛ 1 0 ⎞ then by R12 we get elementary matrix E12 = ⎛ 0
⎜ ⎟ ⎜
1⎞
⎟ .
⎜ ⎟ ⎜ 0⎟
⎝0 1⎠ ⎝1 ⎠
Now E12. A = ⎛ 0
⎜
1⎞
⎟
⎛3
⎜
4⎞=
⎟
⎛8
⎜
9⎞
⎟
⎜1 0⎟ ⎜8 9⎟ ⎜3 4⎟
⎝ ⎠ ⎝ ⎠ ⎝ ⎠
Which is same as seen in (1).
2. The matrix obtained by applying a column operation on a given matrix is same as the matrix obtained
by post-multiplication of the given matrix by the corresponding elementary matrix.
3. The effect of a row operation on a product of two matrices is equivalent to the effect of the same row
operation applied only to pre-factor.
4. The effect of a column operation on a product of two matrices is equivalent to the effect of the same
column operation applied only to the post-factor.
SYSTEM OF LINEAR EQUATIONS
Consider the following system of m linear equations in n unknown x1, x2,….xn.
a11x 1 + a12 x 2 + ..... − a1n x n = b1 ⎫
⎪
a 21x 1 + a 22 x 2 + ..... − a 2n x n = b 2 ⎪
⎪
.......... .......... .......... .......... ............... ⎬
.......... .......... .......... .......... .................⎪
⎪
a m1x 1 + a m2 x 2 + ..... − a mn x n = b m ⎪ ⎭
The above system, of equations can be written as AX = B.
⎛ a11 a12 ....... a1n ⎞ ⎛ x1 ⎞ ⎛ b1 ⎞
⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎜ a 21 a12 ....... a1n ⎟ ⎜ x2 ⎟ ⎜ b2 ⎟
⎜ .... ...... ....... ...... ⎟ ⎜ ..... ⎟ = ⎜ ..... ⎟
⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎜a a m2 ....... a mn ⎟ ⎜x ⎟ ⎜b ⎟
⎝ m1 ⎠ ⎝ n⎠ ⎝ n⎠
If B = 0 i.e., b1 = b2 = b3….bn = 0, then the system of equations is called homogeneous
If B ≠ 0, then the system of equations is non-homogeneous.
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SOLUTION OF A NON-HOMOGENEOUS SYSTEM OF LINEAR EQUATIONS
There are three methods of solving a system of linear equations:
(i) Matrix method
(ii) Rank method
(iii) Determinant method (Cramer's Rule)
Matrix method: For the system of m-linear equations in n unknowns (variables) represented by AX = B.
(i) If |A| ≠ 0, then the system is consistent and has a unique solution given by X = A–1B.
(ii) If |A| = 0 and (adj A) B = 0, then the system is consistent and has infinitely many solutions.
(iii) If |A| = 0 and (adj A) B ≠ 0, then the system is inconsistent (no solution).
Rank method: For the system of m-linear equations in n unknowns (variables) represented by AX = B.
If m > n, then
(i) If r (A) = r (A : B) = n, then system of linear equations has a unique solution.
(ii) If r (A) = r (A : B) = r < n, then system of linear equations is consistent and has infinite number of
solutions. In fact, in this case (n – r) variables can be assigned arbitrary values.
(iii) If r (A) ≠ r (A : B), then the system of linear equations is inconsistent i.e. it has no solution.
If m < n and r (A) = r (A : B) = r, then there are infinite number of solutions.
Determinant method: For a system of 3 simultaneous linear equations in three unknowns.
(i) If D ≠ 0, then the given system of equations is consistent and has a unique solution given by
D1 D D
x= , y = 2 and z = 3
D' D D
(ii) If D = 0 and D1 = D2 = D3 = 0, then the given system of equations is consistent with infinitely
many solutions.
(iii) If D = 0 and at least one of the determinants D1, D2, D3 is non-zero, then the given system of
equations is inconsistent.
SOLUTION OF A HOMOGENEOUS SYSTEM OF LINEAR EQUATIONS
A homogeneous system of linear equations, AX = O is never inconsistent, as it always have a trivial solution
i.e. X = 0.
Matrix method:
Step I: If |A| ≠ 0, then “the system is consistent with unique solution x = y = z = 0.” (trivial solution)
Step II: If |A| = 0, then system of equations has infinitely many solutions.
To find these solutions proceed as follows:
Put z = k (any real number) and solve any two equations for x and y in terms of k. The values of
x and y so obtained with z = k give a solution of the given system of equations.
Rank method:
If r (A) = n = number of variables, then AX = O has a unique solution X = 0 i.e. x1 = x2 = … = 0. (trivial solution)
If r (A) = r < n (= number of variables), then the system of equations has an infinite number of solutions.
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EXAMPLE
⎛1 2 3 ⎞
⎜ ⎟
1. The rank of the matrix A = ⎜ 2 3 4 ⎟ is :
⎜ 4 10 18 ⎟
⎝ ⎠
(a) 0 (b) 1 (c) 2 (d) 3
⎛1 2 3 ⎞
⎜ ⎟
Sol. Let A = ⎜ 2 3 4 ⎟ Therefore
⎜ 4 10 18 ⎟
⎝ ⎠
⎛1 2 3 ⎞
⎜ ⎟
A = ⎜ 2 3 4 ⎟ = 1 (54 – 40) – 2(36 – 16) + 3 (20 – 12) = – 2 ≠ 0.
⎜ 4 10 18 ⎟
⎝ ⎠
Therefore p (A) ≥ 3…..(i). The matrix A does not posses any minor of order 4.
Therefore p(A) ≥ 3 …..(ii) From (i) and (ii) p (A) = 3. Therefore the correct answer is (d)
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