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.
MATRICES maths project.pptxsgdhdghdgf gr to f HR fpremkumar24914
Dfgvvvgggggggdcdfggsggshhdbdybfhfbfhbfbfhfbfhbfbfhfhfjfhfhhfhfhgththththhththththththhhfjfbrhhrbtht me for the information ℹ️ℹ️ and contact information ℹ️ℹ️ and grant you are you 💕💕💕 I don't have main roll no data for the same and the information about it and grant you are not coming to the parent or no data and contact information about the information of the elite year 2 years blood group of the elite year 2 years of the elite of nobody will give me when the parent is not 🚫 to come in detail and t be a consideration and grant you the parent or not feeling well as well as the elite of the parent iam in the parent iam in the parent iam not 🚭🚭 I will kick u in the shadow and the information about the information of the elite and the information of nobody in a sec and grant me a call 🤙🤙 and grant you are you still he is a consideration of my life 🧬🧬 I don't have main places to visit in the shadow of the elite year 2 years of life and the information about it but na vit d I don't have any other questions for you are not coming school year blood test 😔 😔 for jee mains ka I will be taking leave for the information of nobody will give you a sec i will be able and grant me a day or no data for the information about it but I don't have main places to be taking leave for jee to be able to be a sec i will confirm with you are not coming school year 2 days I will be able and grant you to talk to you and contact number of the parent is a consideration and the information about you to be taking the information of nobody will give me when I am in a sec to you and grant you to talk about the parent is not coming to talk about it and I am in a consideration of the day of life and grant me leave taken to talk to you sir and grant me a call 🤙 I am in a sec to you and grant you are not coming school year blood test postion and grant you are not coming school year blood test postion and grant you are not coming school year blood test postion and grant you are not coming school year blood test postion and grant you are not coming school year blood test postion and grant you are not coming school year blood test postion and grant you are not coming school year blood test and grant you are not coming school today my future is not coming to school today my future life and grant me leave taken to the parent or no data for the information of nobody will give me a sec to you sir and the parent iam waiting on the parent iam in detail and grant you are not coming school year 2 years blood test postion and contact information about the information of nobody is okay va va va va va for the information of your work and grant you the same and grant you the same and grant me a call me you will be able to attend a sec to you sir for your work is not coming school year blood test postion and grant you the same to talk to you sir for the same yyrty and grant you are not coming school year blood test postion for the same yyrty and grant you are not available for jee class
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.
MATRICES maths project.pptxsgdhdghdgf gr to f HR fpremkumar24914
Dfgvvvgggggggdcdfggsggshhdbdybfhfbfhbfbfhfbfhbfbfhfhfjfhfhhfhfhgththththhththththththhhfjfbrhhrbtht me for the information ℹ️ℹ️ and contact information ℹ️ℹ️ and grant you are you 💕💕💕 I don't have main roll no data for the same and the information about it and grant you are not coming to the parent or no data and contact information about the information of the elite year 2 years blood group of the elite year 2 years of the elite of nobody will give me when the parent is not 🚫 to come in detail and t be a consideration and grant you the parent or not feeling well as well as the elite of the parent iam in the parent iam in the parent iam not 🚭🚭 I will kick u in the shadow and the information about the information of the elite and the information of nobody in a sec and grant me a call 🤙🤙 and grant you are you still he is a consideration of my life 🧬🧬 I don't have main places to visit in the shadow of the elite year 2 years of life and the information about it but na vit d I don't have any other questions for you are not coming school year blood test 😔 😔 for jee mains ka I will be taking leave for the information of nobody will give you a sec i will be able and grant me a day or no data for the information about it but I don't have main places to be taking leave for jee to be able to be a sec i will confirm with you are not coming school year 2 days I will be able and grant you to talk to you and contact number of the parent is a consideration and the information about you to be taking the information of nobody will give me when I am in a sec to you and grant you to talk about the parent is not coming to talk about it and I am in a consideration of the day of life and grant me leave taken to talk to you sir and grant me a call 🤙 I am in a sec to you and grant you are not coming school year blood test postion and grant you are not coming school year blood test postion and grant you are not coming school year blood test postion and grant you are not coming school year blood test postion and grant you are not coming school year blood test postion and grant you are not coming school year blood test postion and grant you are not coming school year blood test and grant you are not coming school today my future is not coming to school today my future life and grant me leave taken to the parent or no data for the information of nobody will give me a sec to you sir and the parent iam waiting on the parent iam in detail and grant you are not coming school year 2 years blood test postion and contact information about the information of nobody is okay va va va va va for the information of your work and grant you the same and grant you the same and grant me a call me you will be able to attend a sec to you sir for your work is not coming school year blood test postion and grant you the same to talk to you sir for the same yyrty and grant you are not coming school year blood test postion for the same yyrty and grant you are not available for jee class
Matrix theory" redirects here. For the physics topic, see Matrix string theory.
An m × n matrix: the m rows are horizontal and the n columns are vertical. Each element of a matrix is often denoted by a variable with two subscripts. For example, a2,1 represents the element at the second row and first column of the matrix.
In mathematics, a matrix (plural matrices) is a rectangular array[1] (see irregular matrix) of numbers, symbols, or expressions, arranged in rows and columns.[2][3] For example, the dimension of the matrix below is 2 × 3 (read "two by three"), because there are two rows and three columns:
{\displaystyle {\begin{bmatrix}1&9&-13\\20&5&-6\end{bmatrix}}.}{\displaystyle {\begin{bmatrix}1&9&-13\\20&5&-6\end{bmatrix}}.}
Provided that they have the same size (each matrix has the same number of rows and the same number of columns as the other), two matrices can be added or subtracted element by element (see conformable matrix). The rule for matrix multiplication, however, is that two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second (i.e., the inner dimensions are the same, n for an (m×n)-matrix times an (n×p)-matrix, resulting in an (m×p)-matrix). There is no product the other way round, a first hint that matrix multiplication is not commutative. Any matrix can be multiplied element-wise by a scalar from its associated field.
This presentation describes Matrices and Determinants in detail including all the relevant definitions with examples, various concepts and the practice problems.
Classification Matrices execute measurements, for example, Log-Loss, Accuracy, AUC(Area under Curve), and so on. Another case of metric for assessment of AI calculations is exactness, review, which can be utilized for arranging calculations principally utilized via web indexes.
Matrix is the structure squares of information science. They show up in different symbols across dialects. From Numpy clusters in Python to data frames in R, to lattices in MATLAB. The Matrix in its most essential structure is an assortment of numbers masterminded in a rectangular or cluster like the style
Why Fourier Transform
General Properties & Symmetry relations
Formula and steps
magnitude and phase spectra
Convergence Condition
mean-square convergence
Gibbs phenomenon
Direct Delta
Energy Density Spectrum
. Types of Modulation(Analog)
Phase-Frequency Relationships
FM and PM basics
Frequency deviation
MODULATION INDEX
Classification of FM
Narrow Band FM (NBFM)
generating a narrowband FM signal.
Wide Band FM (WBFM).
Carson’s Rule
Generation of WBFM
Average Power
FM BANDWIDTH
Comparing Frequency Modulation to Phase Modulation
ROOT-LOCUS METHOD, Determine the root loci on the real axis /the asymptotes o...Waqas Afzal
Angle and Magnitude Conditions
Example of Root Locus
Steps
constructing a root-locus plot is to locate the open-loop poles and zeros in s-plane.
Determine the root loci on the real axis
Determine the asymptotes of the root loci
Determine the breakaway point.
Closed loop stability via root locus
time domain analysis, Rise Time, Delay time, Damping Ratio, Overshoot, Settli...Waqas Afzal
Time Response- Transient, Steady State
Standard Test Signals- U(t), S(t), R(t)
Analysis of First order system - for Step input
Analysis of second order system -for Step input
Time Response Specifications- Rise Time, Delay time, Damping Ratio, Overshoot, Settling Time
Calculations
state space representation,State Space Model Controllability and Observabilit...Waqas Afzal
State Variables of a Dynamical System
State Variable Equation
Why State space approach
Block Diagram Representation Of State Space Model
Controllability and Observability
Derive Transfer Function from State Space Equation
Time Response and State Transition Matrix
Eigen Value
introduction to modeling, Types of Models, Classification of mathematical mod...Waqas Afzal
Types of Systems
Ways to study system
Model
Types of Models
Why Mathematical Model
Classification of mathematical models
Black box, white box, Gray box
Lumped systems
Dynamic Systems
Simulation
laplace transform and inverse laplace, properties, Inverse Laplace Calculatio...Waqas Afzal
Laplace Transform
-Proof of common function
-properties
-Initial Value and Final Value Problems
Inverse Laplace Calculations
-by identification
-Partial fraction
Solution of Ordinary differential using Laplace and inverse Laplace
Transfer Function, Concepts of stability(critical, Absolute & Relative) Poles...Waqas Afzal
Transfer Function
The Order of Control Systems
Concepts of stability(critical, Absolute & Relative)
Poles, Zeros
Stability calculation
BIBO stability
Transient Response Characteristics
Signal Flow Graph, SFG and Mason Gain Formula, Example solved with Masson Gai...Waqas Afzal
Basic Properties of SFG
Definitions of SFG Terms
SFG Algebra
Relation between SFG and block diagram
Mason Gain Formula
Example solved with Masson Gain Formula
automatic control, Basic Definitions, Classification of Control systems, Requ...Waqas Afzal
Why automatic controls is required
2. Process Variables
controlled variable, manipulated variable
3. Functions of Automatic Control
Measurement
Comparison
Computation
Correction
4.Basic Definitions
System, Plant, Process, Controller, input, output, disturbance
5. Classification of Control systems
Natural, Manmade & Automatic control system
Open-Loop, Close-Loop control System
Linear Vs Nonlinear System
Time invariant vs Time variant
Continuous Data Vs Discrete Data System
Deterministic vs Stochastic System
6. Requirements of an ideal Control system
Accuracy, Sensitivity, noise, Bandwidth, Speed, Oscillations
Vaccine management system project report documentation..pdfKamal Acharya
The Division of Vaccine and Immunization is facing increasing difficulty monitoring vaccines and other commodities distribution once they have been distributed from the national stores. With the introduction of new vaccines, more challenges have been anticipated with this additions posing serious threat to the already over strained vaccine supply chain system in Kenya.
Sachpazis:Terzaghi Bearing Capacity Estimation in simple terms with Calculati...Dr.Costas Sachpazis
Terzaghi's soil bearing capacity theory, developed by Karl Terzaghi, is a fundamental principle in geotechnical engineering used to determine the bearing capacity of shallow foundations. This theory provides a method to calculate the ultimate bearing capacity of soil, which is the maximum load per unit area that the soil can support without undergoing shear failure. The Calculation HTML Code included.
Event Management System Vb Net Project Report.pdfKamal Acharya
In present era, the scopes of information technology growing with a very fast .We do not see any are untouched from this industry. The scope of information technology has become wider includes: Business and industry. Household Business, Communication, Education, Entertainment, Science, Medicine, Engineering, Distance Learning, Weather Forecasting. Carrier Searching and so on.
My project named “Event Management System” is software that store and maintained all events coordinated in college. It also helpful to print related reports. My project will help to record the events coordinated by faculties with their Name, Event subject, date & details in an efficient & effective ways.
In my system we have to make a system by which a user can record all events coordinated by a particular faculty. In our proposed system some more featured are added which differs it from the existing system such as security.
Student information management system project report ii.pdfKamal Acharya
Our project explains about the student management. This project mainly explains the various actions related to student details. This project shows some ease in adding, editing and deleting the student details. It also provides a less time consuming process for viewing, adding, editing and deleting the marks of the students.
Water scarcity is the lack of fresh water resources to meet the standard water demand. There are two type of water scarcity. One is physical. The other is economic water scarcity.
Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
Saudi Arabia stands as a titan in the global energy landscape, renowned for its abundant oil and gas resources. It's the largest exporter of petroleum and holds some of the world's most significant reserves. Let's delve into the top 10 oil and gas projects shaping Saudi Arabia's energy future in 2024.
Immunizing Image Classifiers Against Localized Adversary Attacksgerogepatton
This paper addresses the vulnerability of deep learning models, particularly convolutional neural networks
(CNN)s, to adversarial attacks and presents a proactive training technique designed to counter them. We
introduce a novel volumization algorithm, which transforms 2D images into 3D volumetric representations.
When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
the immunity of models against localized universal attacks by up to 40%. We evaluate our proposed approach
using contemporary CNN architectures and the modified Canadian Institute for Advanced Research (CIFAR-10
and CIFAR-100) and ImageNet Large Scale Visual Recognition Challenge (ILSVRC12) datasets, showcasing
accuracy improvements over previous techniques. The results indicate that the combination of the volumetric
input and curriculum learning holds significant promise for mitigating adversarial attacks without necessitating
adversary training.
Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
In this month's edition, along with this month's industry news to celebrate the 13 years since the group was created we have articles including
A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
Quality defects in TMT Bars, Possible causes and Potential Solutions.PrashantGoswami42
Maintaining high-quality standards in the production of TMT bars is crucial for ensuring structural integrity in construction. Addressing common defects through careful monitoring, standardized processes, and advanced technology can significantly improve the quality of TMT bars. Continuous training and adherence to quality control measures will also play a pivotal role in minimizing these defects.
2. Matrices - Introduction
Matrix algebra has at least two advantages:
•Reduces complicated systems of equations to simple
expressions
•Adaptable to systematic method of mathematical treatment
and well suited to computers
Definition:
A matrix is a set or group of numbers arranged in a square or
rectangular array enclosed by two brackets
1
1
0
3
2
4
d
c
b
a
3. Matrices - Introduction
Properties:
•A specified number of rows and a specified number of
columns
•Two numbers (rows x columns) describe the dimensions
or size of the matrix.
Examples:
3x3 matrix
2x4 matrix
1x2 matrix
3
3
3
5
1
4
4
2
1
2
3
3
3
0
1
0
1
1
1
4. Matrices - Introduction
TYPES OF MATRICES
1. Column matrix or vector:
The number of rows may be any integer but the number of
columns is always 1
2
4
1
3
1
1
21
11
m
a
a
a
5. Matrices - Introduction
TYPES OF MATRICES
2. Row matrix or vector
Any number of columns but only one row
6
1
1
2
5
3
0
n
a
a
a
a 1
13
12
11
6. Matrices - Introduction
TYPES OF MATRICES
3. Rectangular matrix
Contains more than one element and number of rows is not
equal to the number of columns
6
7
7
7
7
3
1
1
0
3
3
0
2
0
0
1
1
1
n
m
7. Matrices - Introduction
TYPES OF MATRICES
4. Square matrix
The number of rows is equal to the number of columns
(a square matrix A has an order of m)
0
3
1
1
1
6
6
0
9
9
1
1
1
m x m
The principal or main diagonal of a square matrix is composed of all
elements aij for which i=j
8. Matrices - Introduction
TYPES OF MATRICES
5. Diagonal matrix
A square matrix where all the elements are zero except those on
the main diagonal
1
0
0
0
2
0
0
0
1
9
0
0
0
0
5
0
0
0
0
3
0
0
0
0
3
i.e. aij =0 for all i = j
aij = 0 for some or all i = j
9. Matrices - Introduction
TYPES OF MATRICES
6. Unit or Identity matrix - I
A diagonal matrix with ones on the main diagonal
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
1
0
0
1
i.e. aij =0 for all i = j
aij = 1 for some or all i = j
ij
ij
a
a
0
0
10. Matrices - Introduction
TYPES OF MATRICES
7. Null (zero) matrix - 0
All elements in the matrix are zero
0
0
0
0
0
0
0
0
0
0
0
0
0
ij
a For all i,j
11. Matrices - Introduction
TYPES OF MATRICES
8. Triangular matrix
A square matrix whose elements above or below the main
diagonal are all zero
3
2
5
0
1
2
0
0
1
3
2
5
0
1
2
0
0
1
3
0
0
6
1
0
9
8
1
12. Matrices - Introduction
TYPES OF MATRICES
8a. Upper triangular matrix
A square matrix whose elements below the main
diagonal are all zero
i.e. aij = 0 for all i > j
3
0
0
8
1
0
7
8
1
3
0
0
0
8
7
0
0
4
7
1
0
4
4
7
1
ij
ij
ij
ij
ij
ij
a
a
a
a
a
a
0
0
0
13. Matrices - Introduction
TYPES OF MATRICES
A square matrix whose elements above the main diagonal are all
zero
8b. Lower triangular matrix
i.e. aij = 0 for all i < j
3
2
5
0
1
2
0
0
1
ij
ij
ij
ij
ij
ij
a
a
a
a
a
a
0
0
0
14. Matrices – Introduction
TYPES OF MATRICES
9. Scalar matrix
A diagonal matrix whose main diagonal elements are
equal to the same scalar
A scalar is defined as a single number or constant
1
0
0
0
1
0
0
0
1
6
0
0
0
0
6
0
0
0
0
6
0
0
0
0
6
i.e. aij = 0 for all i = j
aij = a for all i = j
ij
ij
ij
a
a
a
0
0
0
0
0
0
15. Matrices - Operations
EQUALITY OF MATRICES
Two matrices are said to be equal only when all
corresponding elements are equal
Therefore their size or dimensions are equal as well
3
2
5
0
1
2
0
0
1
3
2
5
0
1
2
0
0
1
A = B = A = B
16. Matrices - Operations
Some properties of equality:
•IIf A = B, then B = A for all A and B
•IIf A = B, and B = C, then A = C for all A, B and C
3
2
5
0
1
2
0
0
1
A = B =
33
32
31
23
22
21
13
12
11
b
b
b
b
b
b
b
b
b
If A = B then ij
ij b
a
17. Matrices - Operations
ADDITION AND SUBTRACTION OF MATRICES
The sum or difference of two matrices, A and B of the same
size yields a matrix C of the same size
ij
ij
ij b
a
c
Matrices of different sizes cannot be added or subtracted
18. Matrices - Operations
SCALAR MULTIPLICATION OF MATRICES
Matrices can be multiplied by a scalar (constant or single
element)
Let k be a scalar quantity; then
kA = Ak
Ex. If k=4 and
1
4
3
2
1
2
1
3
A
20. Matrices - Operations
MULTIPLICATION OF MATRICES
The product of two matrices is another matrix
Two matrices A and B must be conformable for multiplication to
be possible
i.e. the number of columns of A must equal the number of rows
of B
Example.
A x B = C
(1x3) (3x1) (1x1)
21. Matrices - Operations
B x A = Not possible!
(2x1) (4x2)
A x B = Not possible!
(6x2) (6x3)
Example
A x B = C
(2x3) (3x2) (2x2)
24. Matrices - Operations
Assuming that matrices A, B and C are conformable for
the operations indicated, the following are true:
1. AI = IA = A
2. A(BC) = (AB)C = ABC - (associative law)
3. A(B+C) = AB + AC - (first distributive law)
4. (A+B)C = AC + BC - (second distributive law)
Caution!
1. AB not generally equal to BA, BA may not be conformable
2. If AB = 0, neither A nor B necessarily = 0
3. If AB = AC, B not necessarily = C
25. Matrices - Operations
If AB = 0, neither A nor B necessarily = 0
0
0
0
0
3
2
3
2
0
0
1
1
26. Matrices - Operations
To transpose:
Interchange rows and columns
The dimensions of AT are the reverse of the dimensions of A
1
3
5
7
4
2
3
2A
A
1
7
3
4
5
2
2
3
T
T
A
A
2 x 3
3 x 2
28. Matrices - Operations
SYMMETRIC MATRICES
A Square matrix is symmetric if it is equal to its
transpose:
A = AT
d
b
b
a
A
d
b
b
a
A
T
29. Matrices - Operations
DETERMINANT OF A MATRIX
To compute the inverse of a matrix, the determinant is required
Each square matrix A has a unit scalar value called the
determinant of A, denoted by det A or |A|
5
6
2
1
5
6
2
1
A
A
If
then
30. Matrices - Operations
If A = [A] is a single element (1x1), then the determinant is
defined as the value of the element
Then |A| =det A = a11
If A is (n x n), its determinant may be defined in terms of order
(n-1) or less.
31. Matrices - Operations
MINORS
If A is an n x n matrix and one row and one column are deleted,
the resulting matrix is an (n-1) x (n-1) submatrix of A.
The determinant of such a submatrix is called a minor of A and
is designated by mij , where i and j correspond to the deleted
row and column, respectively.
mij is the minor of the element aij in A.
33. Matrices - Operations
Therefore the minor of a12 is:
And the minor for a13 is:
33
31
23
21
12
a
a
a
a
m
32
31
22
21
13
a
a
a
a
m
34. Matrices - Operations
COFACTORS
The cofactor Cij of an element aij is defined as:
ij
j
i
ij m
C
)
1
(
When the sum of a row number i and column j is even, cij = mij and
when i+j is odd, cij =-mij
13
13
3
1
13
12
12
2
1
12
11
11
1
1
11
)
1
(
)
3
,
1
(
)
1
(
)
2
,
1
(
)
1
(
)
1
,
1
(
m
m
j
i
c
m
m
j
i
c
m
m
j
i
c
35. Matrices - Operations
DETERMINANTS CONTINUED
The determinant of an n x n matrix A can now be defined as
n
nc
a
c
a
c
a
A
A 1
1
12
12
11
11
det
The determinant of A is therefore the sum of the products of the
elements of the first row of A and their corresponding cofactors.
(It is possible to define |A| in terms of any other row or column
but for simplicity, the first row only is used)
36. Matrices - Operations
Therefore the 2 x 2 matrix :
22
21
12
11
a
a
a
a
A
Has cofactors :
22
22
11
11 a
a
m
c
And:
21
21
12
12 a
a
m
c
And the determinant of A is:
21
12
22
11
12
12
11
11 a
a
a
a
c
a
c
a
A
38. Matrices - Operations
For a 3 x 3 matrix:
33
32
31
23
22
21
13
12
11
a
a
a
a
a
a
a
a
a
A
The cofactors of the first row are:
31
22
32
21
32
31
22
21
13
31
23
33
21
33
31
23
21
12
32
23
33
22
33
32
23
22
11
)
(
a
a
a
a
a
a
a
a
c
a
a
a
a
a
a
a
a
c
a
a
a
a
a
a
a
a
c
39. Matrices - Operations
The determinant of a matrix A is:
21
12
22
11
12
12
11
11 a
a
a
a
c
a
c
a
A
Which by substituting for the cofactors in this case is:
)
(
)
(
)
( 31
22
32
21
13
31
23
33
21
12
32
23
33
22
11 a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
A
40. Matrices - Operations
Example 2:
1
0
1
3
2
0
1
0
1
A
4
)
2
0
)(
1
(
)
3
0
)(
0
(
)
0
2
)(
1
(
A
)
(
)
(
)
( 31
22
32
21
13
31
23
33
21
12
32
23
33
22
11 a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
A
41. Matrices - Operations
ADJOINT MATRICES
A cofactor matrix C of a matrix A is the square matrix of the same
order as A in which each element aij is replaced by its cofactor cij .
Example:
4
3
2
1
A
1
2
3
4
C
If
The cofactor C of A is
42. Matrices - Operations
The adjoint matrix of A, denoted by adj A, is the transpose of its
cofactor matrix
T
C
adjA
It can be shown that:
A(adj A) = (adjA) A = |A| I
Example:
1
3
2
4
10
)
3
)(
2
(
)
4
)(
1
(
4
3
2
1
T
C
adjA
A
A
48. Matrices - Operations
The result can be checked using
AA-1 = A-1 A = I
The determinant of a matrix must not be zero for the inverse to
exist as there will not be a solution
Nonsingular matrices have non-zero determinants
Singular matrices have zero determinants
49. Simple 2 x 2 case
So that for a 2 x 2 matrix the inverse can be constructed
in a simple fashion as
a
c
b
d
A
A
a
A
c
A
b
A
d
1
•Exchange elements of main diagonal
•Change sign in elements off main diagonal
•Divide resulting matrix by the determinant
z
y
x
w
A 1
51. Ch5_51
5.1 Eigenvalues and Eigenvectors
Definition
Let A be an n n matrix. A scalar is called an eigenvalue of A
if there exists a nonzero vector x in Rn such that
Ax = x.
The vector x is called an eigenvector corresponding to .
Figure 5.1
52. Ch5_52
Computation of Eigenvalues and
Eigenvectors
Let A be an n n matrix with eigenvalue and corresponding
eigenvector x. Thus Ax = x. This equation may be written
Ax – x = 0
given
(A – In)x = 0
Solving the equation |A – In| = 0 for leads to all the eigenvalues
of A.
On expending the determinant |A – In|, we get a polynomial in .
This polynomial is called the characteristic polynomial of A.
The equation |A – In| = 0 is called the characteristic equation of
A.
53. Find the eigenvalues and eigenvectors of the matrix
5
3
6
4
A
Let us first derive the characteristic polynomial of A.
We get
Solution
5
3
6
4
1
0
0
1
5
3
6
4
2
I
A
2
18
)
5
)(
4
( 2
2
I
A
We now solve the characteristic equation of A.
The eigenvalues of A are 2 and –1.
The corresponding eigenvectors are found by using these values
of in the equation(A – I2)x = 0. There are many eigenvectors
corresponding to each eigenvalue.
1
or
2
0
)
1
)(
2
(
0
2
2
54. • For = 2
We solve the equation (A – 2I2)x = 0 for x.
The matrix (A – 2I2) is obtained by subtracting 2 from the
diagonal elements of A. We get
0
2
1
3
3
6
6
x
x
This leads to the system of equations
giving x1 = –x2. The solutions to this system of equations are
x1 = –r, x2 = r, where r is a scalar. Thus the eigenvectors of A
corresponding to = 2 are nonzero vectors of the form
0
3
3
0
6
6
2
1
2
1
x
x
x
x
1
1 2
2
1 1
v
1 1
x
x r
x
55. • For = –1
We solve the equation (A + 1I2)x = 0 for x.
The matrix (A + 1I2) is obtained by adding 1 to the diagonal
elements of A. We get
0
2
1
6
3
6
3
x
x
This leads to the system of equations
Thus x1 = –2x2. The solutions to this system of equations are
x1 = –2s and x2 = s, where s is a scalar. Thus the eigenvectors
of A corresponding to = –1 are nonzero vectors of the form
0
6
3
0
6
3
2
1
2
1
x
x
x
x
1
2 2
2
2 2
v
1 1
x
x s
x
56. Find the eigenvalues and eigenvectors of the matrix
2
2
2
2
5
4
2
4
5
A
The matrix A – I3 is obtained by subtracting from
the diagonal elements of A.Thus
Solution
2
2
2
2
5
4
2
4
5
3
I
A
2
2
2
2
5
4
0
1
1
2
2
2
2
5
4
2
4
5
3
I
A
The characteristic polynomial of A is |A – I3|. Using row and
column operations to simplify determinants, we get
60. RANK OF A MATRIX
Let A be any m n matrix. Then A consists of n column
vectors a₁, a₂ ,....,a, which are m-vectors.
DEFINTION:
The rank of A is the maximal number of linearly
independent column vectors in A, i.e. the maximal
number of linearly independent vectors among {a₁,
a₂,......, a}.
If A = 0, then the rank of A is 0.
We write rk(A) for the rank of A. Note that we may
compute the rank of any matrix-square or not
61. Let us see how to compute 22 matrix:
EXAMPLE :
rk ( A) 2 if det A ad bc 0,since both columnvectors are
independent in this case.
The rank of a 22 matrix A =
a b
is given by
c d
rk(A) = 1 if det(A) = 0 but A 0 =
0 0
,since both column vectors
0 0
RANK OF 22 MATRIX
are not linearly independent, but there is a single column vector that is
linearly independent (i.e. non-zero).
rk(A) = 0 if A = 0
How do we compute rk(A) of m x n matrix?
63. SOLUTION:
We use elementary row operations:
2 4 2 4
1 0 2 1 1 0 2 1
A= 0 2 0 2
0 2 2 1 0 0 2 1
Since the echelon form has pivots in the first three columns,
A has rank, rk(A) = 3. The first three columns of A are linearly independent.