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.
GATE Engineering Maths : System of Linear EquationsParthDave57
This document discusses linear systems of equations and their solutions. It defines linear equations and systems, and describes how to write a system of equations in matrix form. It then explains how to use Gaussian elimination to put the augmented matrix of a linear system into row echelon form and thus solve for the variables. The document also defines the rank of a matrix, and discusses when systems have unique, multiple, or no solutions based on the rank. It concludes by defining linear dependence and independence of vectors.
This document provides an overview of matrix algebra concepts including:
- Matrices are rectangular arrays of numbers that let us express large amounts of data in an organized form. Common types include vectors, square matrices, and rectangular matrices.
- Matrices can be added or multiplied by scalars if they are the same size. Matrix multiplication follows specific rules such as being non-commutative.
- Special types of matrices include symmetric, skew-symmetric, upper/lower triangular, diagonal, and identity matrices.
- Determinants are values that can be calculated for square matrices and represent a type of scaling factor.
A matrix is a rectangular arrangement of numbers organized in rows and columns. The order of a matrix refers to the number of rows and columns it contains. Entries are the individual numbers within the matrix. Basic matrix operations include addition, subtraction, scalar multiplication, and multiplication. To add or subtract matrices, they must be the same order, while scalar multiplication multiplies each entry of the matrix by the scalar.
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.
All mathematical truths are relative and conditional. Determinants are numbers associated with square matrices that represent the signed area or volume of geometric objects defined by the matrices. This document defines properties of determinants, such as how they are calculated for different sized matrices, how operations on the rows and columns of a matrix affect its determinant, and properties like determinants of diagonal, symmetric, and skew-symmetric matrices. It also provides examples of using determinants to find equations of lines and the area of triangles.
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.
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
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.
GATE Engineering Maths : System of Linear EquationsParthDave57
This document discusses linear systems of equations and their solutions. It defines linear equations and systems, and describes how to write a system of equations in matrix form. It then explains how to use Gaussian elimination to put the augmented matrix of a linear system into row echelon form and thus solve for the variables. The document also defines the rank of a matrix, and discusses when systems have unique, multiple, or no solutions based on the rank. It concludes by defining linear dependence and independence of vectors.
This document provides an overview of matrix algebra concepts including:
- Matrices are rectangular arrays of numbers that let us express large amounts of data in an organized form. Common types include vectors, square matrices, and rectangular matrices.
- Matrices can be added or multiplied by scalars if they are the same size. Matrix multiplication follows specific rules such as being non-commutative.
- Special types of matrices include symmetric, skew-symmetric, upper/lower triangular, diagonal, and identity matrices.
- Determinants are values that can be calculated for square matrices and represent a type of scaling factor.
A matrix is a rectangular arrangement of numbers organized in rows and columns. The order of a matrix refers to the number of rows and columns it contains. Entries are the individual numbers within the matrix. Basic matrix operations include addition, subtraction, scalar multiplication, and multiplication. To add or subtract matrices, they must be the same order, while scalar multiplication multiplies each entry of the matrix by the scalar.
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.
All mathematical truths are relative and conditional. Determinants are numbers associated with square matrices that represent the signed area or volume of geometric objects defined by the matrices. This document defines properties of determinants, such as how they are calculated for different sized matrices, how operations on the rows and columns of a matrix affect its determinant, and properties like determinants of diagonal, symmetric, and skew-symmetric matrices. It also provides examples of using determinants to find equations of lines and the area of triangles.
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.
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
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 defines matrices and determinants, including examples and types of matrices. It describes how to add, subtract, and multiply matrices, and defines determinants and Cramer's rule. Cramer's rule is used to solve a 3x3 system of equations. The relationship between matrices and determinants is that determinants are uniquely related to square matrices but not vice versa, and determinants are used to calculate inverses.
The document provides an overview of a lecture covering matrices, matrix algebra, vectors, homogeneous coordinates, and transformations in homogeneous coordinates. Key points include: matrices are arrays of numbers; operations on matrices include addition, multiplication by a scalar, and multiplication; vectors can represent points in space as column or row matrices; homogeneous coordinates allow points to be represented by 4D vectors, enabling translations and other transformations to be described by 4x4 matrices. This provides a unified approach for combining multiple transformations.
A matrix is a rectangular array of numbers arranged in rows and columns. The dimensions of a matrix are written as the number of rows x the number of columns. Each individual entry in the matrix is named by its position, using the matrix name and row and column numbers. Matrices can represent systems of equations or points in a plane. Operations on matrices include addition, multiplication by scalars, and dilation of points represented by matrices.
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.
This document discusses basic matrix operations including:
- Defining a matrix as a rectangular arrangement of numbers in rows and columns with an order specified by the number of rows and columns.
- Adding and subtracting matrices requires they have the same order and involves adding or subtracting corresponding entries.
- Multiplying a matrix by a scalar involves multiplying each entry in the matrix by the scalar value.
- Matrix multiplication is not commutative and can only be done if the number of columns in the first matrix equals the number of rows in the second matrix. It involves multiplying entries and summing the products based on their positions.
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.
The document provides definitions and concepts related to matrices and determinants. It begins with definitions of matrices, operations on matrices like transpose and trace. It then discusses row echelon form, elementary row operations, and using matrices to represent systems of linear equations. The document will cover topics like inverse matrices, matrix rank and nullity, polynomials of matrices, properties of determinants, minors and cofactors, and Cramer's rule.
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.
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 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 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
The document discusses determinants and their properties. It defines determinants as scalars associated with square matrices. Determinants can be computed for matrices of any size using expansions by cofactors along rows or columns. The key properties of determinants are that they are unchanged by row operations and that the inverse of a non-singular matrix can be computed using the adjugate matrix divided by the determinant. The document provides examples of computing the determinant, adjugate, and inverse of matrices.
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.
Matrix and its operation (addition, subtraction, multiplication)NirnayMukharjee
This document summarizes matrix operations including addition, subtraction, and multiplication. It defines a matrix as a rectangular arrangement of numbers in rows and columns. Matrix addition and subtraction can only be done on matrices with the same dimensions, by adding or subtracting the corresponding elements. Matrix multiplication involves multiplying the rows of the first matrix with the columns of the second matrix and summing the products to form the elements of the resulting matrix. Examples are provided to illustrate each operation.
This document discusses Gauss-Jordan elimination for solving systems of linear equations. It begins by introducing the three possible cases for solutions: unique solution, no solution, or infinite solutions. It then provides an example of using Gauss-Jordan elimination to solve a 3x3 system. The steps involve transforming the augmented matrix into reduced row echelon form and then reading the solution variables from the final matrix. Applications of solving systems from word problems are also discussed.
This document discusses matrix algebra concepts such as determinants, inverses, eigenvalues, and rank. It provides the following key points:
- The determinant of a square matrix is a number that characterizes properties like singularity. It is defined as the sum of products of the matrix elements.
- Cramer's rule provides a formula for solving systems of linear equations using determinants, but it is only practical for small matrices up to 3x3 or 4x4 due to computational complexity.
- A matrix is singular if its determinant is zero, meaning its rows and columns are linearly dependent. The rank of a matrix is the size of the largest non-singular sub-matrix. A full rank matrix has
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive function. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms.
Properties in perumbakam. The slides show you the highlights of building, that the lofts are suitable for all buyers needs (i.e) built with the purpose of Regular, Compact and with study room.
The Villa Bonne Nouvelle experimented with new ways of working in a digital and collaborative environment. It hosted multidisciplinary project teams from Orange and external startups/freelancers. The goal was to analyze how the work environment impacts employee satisfaction and engagement. Key findings after one year include identifying actions to improve conviviality, space flexibility, and real-time collaboration tools to enable agility. The villa concluded its role as an incubator for new collaborative and managerial practices.
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 defines matrices and determinants, including examples and types of matrices. It describes how to add, subtract, and multiply matrices, and defines determinants and Cramer's rule. Cramer's rule is used to solve a 3x3 system of equations. The relationship between matrices and determinants is that determinants are uniquely related to square matrices but not vice versa, and determinants are used to calculate inverses.
The document provides an overview of a lecture covering matrices, matrix algebra, vectors, homogeneous coordinates, and transformations in homogeneous coordinates. Key points include: matrices are arrays of numbers; operations on matrices include addition, multiplication by a scalar, and multiplication; vectors can represent points in space as column or row matrices; homogeneous coordinates allow points to be represented by 4D vectors, enabling translations and other transformations to be described by 4x4 matrices. This provides a unified approach for combining multiple transformations.
A matrix is a rectangular array of numbers arranged in rows and columns. The dimensions of a matrix are written as the number of rows x the number of columns. Each individual entry in the matrix is named by its position, using the matrix name and row and column numbers. Matrices can represent systems of equations or points in a plane. Operations on matrices include addition, multiplication by scalars, and dilation of points represented by matrices.
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.
This document discusses basic matrix operations including:
- Defining a matrix as a rectangular arrangement of numbers in rows and columns with an order specified by the number of rows and columns.
- Adding and subtracting matrices requires they have the same order and involves adding or subtracting corresponding entries.
- Multiplying a matrix by a scalar involves multiplying each entry in the matrix by the scalar value.
- Matrix multiplication is not commutative and can only be done if the number of columns in the first matrix equals the number of rows in the second matrix. It involves multiplying entries and summing the products based on their positions.
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.
The document provides definitions and concepts related to matrices and determinants. It begins with definitions of matrices, operations on matrices like transpose and trace. It then discusses row echelon form, elementary row operations, and using matrices to represent systems of linear equations. The document will cover topics like inverse matrices, matrix rank and nullity, polynomials of matrices, properties of determinants, minors and cofactors, and Cramer's rule.
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.
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 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 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
The document discusses determinants and their properties. It defines determinants as scalars associated with square matrices. Determinants can be computed for matrices of any size using expansions by cofactors along rows or columns. The key properties of determinants are that they are unchanged by row operations and that the inverse of a non-singular matrix can be computed using the adjugate matrix divided by the determinant. The document provides examples of computing the determinant, adjugate, and inverse of matrices.
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.
Matrix and its operation (addition, subtraction, multiplication)NirnayMukharjee
This document summarizes matrix operations including addition, subtraction, and multiplication. It defines a matrix as a rectangular arrangement of numbers in rows and columns. Matrix addition and subtraction can only be done on matrices with the same dimensions, by adding or subtracting the corresponding elements. Matrix multiplication involves multiplying the rows of the first matrix with the columns of the second matrix and summing the products to form the elements of the resulting matrix. Examples are provided to illustrate each operation.
This document discusses Gauss-Jordan elimination for solving systems of linear equations. It begins by introducing the three possible cases for solutions: unique solution, no solution, or infinite solutions. It then provides an example of using Gauss-Jordan elimination to solve a 3x3 system. The steps involve transforming the augmented matrix into reduced row echelon form and then reading the solution variables from the final matrix. Applications of solving systems from word problems are also discussed.
This document discusses matrix algebra concepts such as determinants, inverses, eigenvalues, and rank. It provides the following key points:
- The determinant of a square matrix is a number that characterizes properties like singularity. It is defined as the sum of products of the matrix elements.
- Cramer's rule provides a formula for solving systems of linear equations using determinants, but it is only practical for small matrices up to 3x3 or 4x4 due to computational complexity.
- A matrix is singular if its determinant is zero, meaning its rows and columns are linearly dependent. The rank of a matrix is the size of the largest non-singular sub-matrix. A full rank matrix has
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive function. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms.
Properties in perumbakam. The slides show you the highlights of building, that the lofts are suitable for all buyers needs (i.e) built with the purpose of Regular, Compact and with study room.
The Villa Bonne Nouvelle experimented with new ways of working in a digital and collaborative environment. It hosted multidisciplinary project teams from Orange and external startups/freelancers. The goal was to analyze how the work environment impacts employee satisfaction and engagement. Key findings after one year include identifying actions to improve conviviality, space flexibility, and real-time collaboration tools to enable agility. The villa concluded its role as an incubator for new collaborative and managerial practices.
The document outlines Lexus's digital media strategy to target middle-aged, high-income individuals. It recommends using Pinterest advertising given that the platform is 80% female, which could help balance Lexus's current male-focused ads. It also suggests display ads on sites like Autotrader and Carfax that the target audience frequents, as well as sponsoring a NASCAR driver to appeal to racing fans who tend to have higher incomes. Key metrics to measure include reach, interaction rates, cost per thousand impressions, clicks, conversion rates, and sales.
Matt Ullrich is an account executive at Hub International in Casper, Wyoming. He works with businesses in the region to maximize their property and casualty commercial insurance portfolios. Previously, Matt worked in the oil industry as a wireline operator and field engineer in the Bakken, Powder River, and DJ Basins, gaining experience he passes on to new hires. Matt has a bachelor's degree in psychology and sociology from Black Hills State University and a second bachelor's degree in business management from the same AACSB accredited school.
This document provides an offering summary for a net lease investment property occupied by US Bank in Lockport, Illinois. US Bank recently signed a 10-year lease extension through 2026 for the 5,250 square foot building on nearly 45,000 square feet of land. The property benefits from its location along a major road near retail tenants like Walmart and Jewel-Osco. US Bank is an investment grade rated tenant, and the lease features annual rent escalations of 2.5% with renewal options. The trade area has a population over 112,000 with above average incomes and a well-educated demographic.
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.
BSC_COMPUTER _SCIENCE_UNIT-3_DISCRETE MATHEMATICSRai University
The document provides information about matrices including:
- Definitions of matrix, order of a matrix, types of matrices like row, column, square, null, identity, diagonal, scalar, and transpose matrices.
- Operations on matrices like addition, subtraction, scalar multiplication and matrix multiplication along with their properties.
- Examples of finding the sum of matrices.
- The document discusses various types of matrices like symmetric, skew-symmetric, singular, equal, negative, orthogonal, upper/lower triangular, trace, idempotent, involuntary, conjugate, unitary, Hermitian, skew-Hermitian matrices and minors of a matrix.
This document discusses determinants and their properties. It begins by introducing determinants as functions that associate a unique number (real or complex) to square matrices. It then provides examples of calculating determinants of matrices of order 1, 2 and 3 by expanding along rows or columns. The key properties discussed are that the determinant remains unchanged under row/column interchange and that the determinant of a scalar multiple of a matrix is the scalar raised to the power of the matrix order times the determinant of the original matrix. Examples are provided to illustrate these concepts.
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.
This document discusses matrices and determinants. It begins by defining a matrix as a rectangular array of numbers or other objects arranged in rows and columns. It then discusses types of matrices, equality of matrices, and algebraic operations on matrices. The document also covers symmetric and skew-symmetric matrices, determinants of different matrix orders, properties of determinants such as how they change when rows/columns are swapped, and the product of determinants.
1. The determinant of a matrix summarizes the whole matrix and can be computed using cofactor expansions along rows or columns.
2. Elementary row operations such as interchanging rows, multiplying a row by a constant, or adding a multiple of one row to another do not change the determinant of a matrix.
3. A matrix is invertible if and only if its determinant is not equal to zero.
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.
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.
The document defines and provides examples of different types of matrices such as square, diagonal, identity, and zero matrices. It also discusses matrix operations like addition, subtraction, scalar multiplication, and matrix multiplication. Key properties of these operations such as commutativity, associativity, and invertibility are covered. Matrix transpose and elementary row operations are also introduced.
For the following matrices, determine a cot of basis vectors for the.pdfeyebolloptics
For the following matrices, determine a cot of basis vectors for the null spaces the column
spaces. A = [3 2 -1 5] A = [2 1 -3 4 0 2 0 -2 8] Let P_n denote the set of all polynomial
functions of degree at most n Let a_o be a feed constant. Explain why H = [p(t) = a_0 + b_x| b
R} is not necessarily a vector subspace of P_x, Are there any values of a_o for which H will be a
subspace? If instead H = {p(t) = a + bt| a, b R}, (i.e., the constant term is allowed to vary over
all real numbers), show that H is a vector subspace of P_n.
Solution
Ans-
A matrix, in general sense, represents a
collection of information stored or arranged
in an orderly fashion. The mathematical
concept of a matrix refers to a set of numbers,
variables or functions ordered in rows and
columns. Such a set then can be defined as a
distinct entity, the matrix, and it can be
manipulated as a whole according to some
basic mathematical rules.
A matrix with 9 elements is shown below.
[
[[
[ ]
]]
]
=
==
=
aaa
=
==
=
253
A
131211
aaa
aaa
232221
333231
819
647
Matrix [A] has 3 rows and 3 columns. Each
element of matrix [A] can be referred to by its
row and column number. For example,
=
==
=a
23
6
A computer monitor with 800 horizontal
pixels and 600 vertical pixels can be viewed as
a matrix of 600 rows and 800 columns.
In order to create an image, each pixel is
filled with an appropriate colour.
ORDER OF A MATRIX
The order of a matrix is defined in terms of
its number of rows and columns.
Order of a matrix = No. of rows
×
××
×
No. of
columns
Matrix [A], therefore, is a matrix of order 3
×
××
×
3.
COLUMN MATRIX
A matrix with only one column is called a
column matrix or column vector.
ROW MATRIX
3
6
4
A matrix with only one row is called a row
matrix or row vector.
[
[[
[ ]
]]
]
653
SQUARE MATRIX
A matrix having the same number of rows
and columns is called a square matrix.
742
942
435
RECTANGULAR MATRIX
A matrix having unequal number of rows and
columns is called a rectangular matrix.
1735
13145
8292
REAL MATRIX
A matrix with all real elements is called a real
matrix
PRINCIPAL DIAGONAL and TRACE
OF A MATRIX
In a square matrix, the diagonal containing
the elements a
11
, a
22
, a
33
, a
44
, ……, a
is called
the principal or main diagonal.
The sum of all elements in the principal
diagonal is called the trace of the matrix.
The principal diagonal of the matrix
742
942
435
nn
is indicated by the dashed box. The trace of
the matrix is 2 + 3 + 9 = 14.
UNIT MATRIX
A square matrix in which all elements of the
principal diagonal are equal to 1 while all
other elements are zero is called the unit
matrix.
001
100
010
ZERO or NULL MATRIX
A matrix whose elements are all equal to zero
is called the null or zero matrix.
000
000
000
DIAGONAL MATRIX
If all elements except the elements of the
principal diagonal of a square matrix are
zero, the matrix is called a diagonal matrix.
002
900
030
RANK OF A MATRIX
The maximum number of linearly
independent rows of a matrix [A] is called
the rank of [A] and i.
This document defines determinants and their properties, and provides methods for calculating determinants of matrices. It discusses:
- The definition and properties of determinants
- Calculating determinants through cofactor expansion or reducing the matrix to row echelon form
- Properties related to determinants of sums, products, inverses and adjoints of matrices
- Using determinants to determine the rank of a matrix
- Calculating specific determinants as examples
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.
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.
The document discusses matrices and determinants. It defines a matrix as a rectangular array of numbers arranged in rows and columns. A system of linear equations can be represented using matrices. Special types of matrices include diagonal matrices, where only the diagonal elements are non-zero, and identity matrices, where the diagonal elements are 1 and all others are 0. Matrices can be added or subtracted if they are the same size by adding or subtracting the corresponding elements. Determinants provide a value for a square matrix and are used in solutions to systems of linear equations.
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This document contains every topic of Matrices and Determinants which is helpful for both college and school students:
Matrices
Types of Matrices
Operations of Matrices
Determinants
Minor of Matrix
Co-factor
Ad joint
Transpose
Inverse of matrix
Linear Equation Matrix Solution
Cramer's Rule
Gauss Jordan Elimination Method
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Determinants
1. DETERMINANTS
1. Every square matrix can be associated to an expression or a
number which is known as its determinant.
i) If A =
𝑎₁₁ 𝑎₁₂
𝑎₂₁ 𝑎₂₂ is a square matrix of order 2 X 2, then its
determinant is denoted by
|A| or,
𝑎₁₁ 𝑎₁₂
𝑎₂₁ 𝑎₂₂ and is defined as a11 a22 – a12 a21.
i.e. |A| =
𝑎₁₁ 𝑎₁₂
𝑎₂₁ 𝑎₂₂ = a11 a22 – a12 a21
ii) If A =
𝑎₁₁ 𝑎₁₂ 𝑎₁₃
𝑎₂₁ 𝑎₂₂ 𝑎₂₃
𝑎₃₁ 𝑎₃₂ 𝑎₃
is a square matrix of order 3 X 3,
𝑎₁₁ 𝑎₁₂ 𝑎₁₃
𝑎₂₁ 𝑎₂₂ 𝑎₂₃
𝑎₃₁ 𝑎₃₂ 𝑎₃
then its determinant is denoted by |A| or,
𝑎₁₁ 𝑎₁₂ 𝑎₁₃
𝑎₂₁ 𝑎₂₂ 𝑎₂₃
𝑎₃₁ 𝑎₃₂ 𝑎₃
and is equal to a11 a22 a33 + a12 a23 a31 + a13 a32 a21 – a11 a23 a32 -
a22 a13 a31 – a12 a21 a33
2. This expression can be arranged in the following form:
𝑎₁₁ 𝑎₁₂ 𝑎₁₃
𝑎₂₁ 𝑎₂₂ 𝑎₂₃
𝑎₃₁ 𝑎₃₂ 𝑎₃
= (-1)1 + 1 a11
𝑎₂₂ 𝑎₂₃
𝑎₃₂ 𝑎₃ + (-1)1 + 2 a12
𝑎₂₂ 𝑎₂₃
𝑎₃₂ 𝑎₃₃
+ (-1)1 + 3 a13
𝑎₂₁ 𝑎₂₂
𝑎₃₁ 𝑎₃₂
This is known as the expansion of |A| along first row.
In fact, |A| can be expanded along any of its rows or columns. In order
to expand |A| along any row or column, we multiply
Example 1: - Evaluate the determinant
D =
2 3 −2
1 2 3
−2 1 −3
by expanding it along first column.
SOLUTION: By using the definition, of expansion along first column, we
obtain
D =
2 3 −2
1 2 3
−2 1 −3
3. D = (-1)1+1 (2)
2 3
1 −3
+ (-1)2+1 (1)
3 −2
1 −3
+ (-1)3+1 (-2)
3 −2
1 −3
D = 2
2 3
1 −3
-
3 −2
1 −3
-2
3 −2
1 −3
D = 2 (-6-3) – (-9+2) -2(9+4) = -18 +7-26 = -37.
NOTE 1: Only square matrices have their determinants. The matrices
which are not square do not have determinants.
NOTE 2: The determinant of a square matrix of order 3 can be
expressed along any row or column.
NOTE 3: If a row or a column of a determinant consists of all zeros, then
the value of the determinant is zero.
There are three rows and three columns in a square matrix of order 3.
PROPERTIES OF DETERMINANTS
We have defined the determinants of a square matrix of order 4 or less.
In fact, these definitions are consequences of the general definition of
the determinant of a square matrix of any order which needs so many
advanced concepts. These concepts are beyond the scope of this book.
Using the said definition and some other advanced concepts we can
prove the following properties. But, the concepts used in the definition
itself are very advanced. Therefore we mention and verify them for a
determinant of a square matrix of order 3.
4. Property 1: let A = [aij] be a square matrix of order n, then the sum of
the product of elements of any row(column) with their cofactors is
always equal to |A| or, det (A).
Property 2: let A = [aij] be a square matrix of order n, then the sum of
the product of elements of any row(column) with the cofactors of the
corresponding elements of some other row (column) is zero.
Property 3: Let A = [aij] be a square matrix of order n, then |A| = |AT|.
Property 4: let A = [aij] be a square matrix of order n(≥2) and let B be a
matrix obtained from A by interchanging any two rows(columns) of A,
then |B| = -|A|.
Conventionally this property is also stated as:
1. If any two rows (columns) of a determinant are interchanged, then
the value of the determinant changes by minus sign only.
Property 5: if any two rows (columns) of a square matrix A = [aij] of
order n (>2) are identical, then its determinant is zero i.e. |A| = 0.
Property 6: Let A = [aij] be a square matrix of order n, and let B be
the matrix obtained from A by multiplying each element of a row
(column) of A by a scalar k, then |B| = k |A|.
5. Property 7: Let A square matrix such that each element of row
(column) of A is expressed as the sum of two or more terms. Then,
the determinant of A can be expressed as the sum of the
determinants of two or more matrices of the same order.