9.3 Solving Systems With Gaussian Eliminationsmiller5
Write the augmented matrix of a system of equations.
Write the system of equations from an augmented matrix.
Perform row operations on a matrix.
Solve a system of linear equations using matrices.
9.3 Solving Systems With Gaussian Eliminationsmiller5
Write the augmented matrix of a system of equations.
Write the system of equations from an augmented matrix.
Perform row operations on a matrix.
Solve a system of linear equations using matrices.
Chapter 5: Determinant
Covered Topics:
5.1 Definition of Determinant
5.2 Expansion of Determinant of order 2X3
5.3 Crammer’s rule to solve simultaneous equations in 3 unknowns
Youtube Link: https://youtu.be/C2qctvyjG7U
Document:
Our Blog Link: http://jjratnani.wordpress.com/
Find the nth term of a sequence
Find the index of a given term of a sequence
Given a geometric series, be able to calculate the nth partial sum
Identify a geometric series as convergent or divergent.
This presentation describes Matrices and Determinants in detail including all the relevant definitions with examples, various concepts and the practice problems.
Chapter 5: Determinant
Covered Topics:
5.1 Definition of Determinant
5.2 Expansion of Determinant of order 2X3
5.3 Crammer’s rule to solve simultaneous equations in 3 unknowns
Youtube Link: https://youtu.be/C2qctvyjG7U
Document:
Our Blog Link: http://jjratnani.wordpress.com/
Find the nth term of a sequence
Find the index of a given term of a sequence
Given a geometric series, be able to calculate the nth partial sum
Identify a geometric series as convergent or divergent.
This presentation describes Matrices and Determinants in detail including all the relevant definitions with examples, various concepts and the practice problems.
MATRICES maths project.pptxsgdhdghdgf gr to f HR fpremkumar24914
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* Model exponential growth and decay
* Use Newton's Law of Cooling
* Use logistic-growth models
* Choose an appropriate model for data
* Express an exponential model in base e
* Construct perpendicular and angle bisectors
* Use bisectors to solve problems
* Identify the circumcenter and incenter of a triangle
* Use triangle segments to solve problems
* Identify, write, and analyze conditional statements
* Write the inverse, converse, and contrapositive of a conditional statement
* Write a counterexample to a fake conjecture
* Find the distance between two points
* Find the midpoint of two given points
* Find the coordinates of an endpoint given one endpoint and a midpoint
* Find the coordinates of a point a fractional distance from one end of a segment
* Connect functions to their graphs
* Graph piecewise-defined functions
* Graph absolute value functions
* Graph greatest-integer functions
* Interpret graphs
* Use the vertical line test to determine a function
* Connect functions to their graphs
* Graph piecewise-defined functions
* Graph absolute value functions
* Graph greatest-integer functions
* Interpret graphs
* Use the vertical line test to determine a function
* Introduce functions and function notation
* Develop skills in constructing and interpreting the graphs of functions
* Learn to apply this knowledge in a variety of situations
* Recognize graphs of common functions.
* Graph functions using vertical and horizontal shifts.
* Graph functions using reflections about the x-axis and the y-axis.
* Graph functions using compressions and stretches.
* Combine transformations.
* Identify intervals on which a function increases, decreases, or is constant
* Use graphs to locate relative maxima or minima
* Test for symmetry
* Identify even or odd functions and recognize their symmetries
* Understand and use piecewise functions
* Solve polynomial equations by factoring
* Solve equations with radicals and check the solutions
* Solve equations with rational exponents
* Solve equations that are quadratic in form
* Solve absolute value equations
* Determine whether a relation or an equation represents a function.
* Evaluate a function.
* Use the vertical line test to identify functions.
* Identify the domain and range of a function from its graph
* Identify intercepts from a function’s graph
* Solve counting problems using the Addition Principle.
* Solve counting problems using the Multiplication Principle.
* Solve counting problems using permutations involving n distinct objects.
* Solve counting problems using combinations.
* Find the number of subsets of a given set.
* Solve counting problems using permutations involving n non-distinct objects.
* Use summation notation.
* Use the formula for the sum of the first n terms of an arithmetic series.
* Use the formula for the sum of the first n terms of a geometric series.
* Use the formula for the sum of an infinite geometric series.
* Solve annuity problems.
* Find the common ratio for a geometric sequence.
* List the terms of a geometric sequence.
* Use a recursive formula for a geometric sequence.
* Use an explicit formula for a geometric sequence.
* Find the common difference for an arithmetic sequence.
* Write terms of an arithmetic sequence.
* Use a recursive formula for an arithmetic sequence.
* Use an explicit formula for an arithmetic sequence.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
Honest Reviews of Tim Han LMA Course Program.pptxtimhan337
Personal development courses are widely available today, with each one promising life-changing outcomes. Tim Han’s Life Mastery Achievers (LMA) Course has drawn a lot of interest. In addition to offering my frank assessment of Success Insider’s LMA Course, this piece examines the course’s effects via a variety of Tim Han LMA course reviews and Success Insider comments.
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
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2. Concepts and Objectives
⚫ Find the sum and difference of two matrices.
⚫ Find scalar multiples of a matrix.
⚫ Find the product of two matrices.
3. Introduction to Matrices
⚫ A matrix (plural matrices) is a rectangular array of
numbers enclosed in brackets. Each number is called an
element of the matrix.
⚫ A row in a matrix is a set of numbers that are aligned
horizontally. A column is a set of numbers that are
aligned vertically.
⚫ We generally use capital letters for the names of
matrices.
⚫ Examples:
1 3 1 2 7
1 2
, 4 0 , 0 5 6
3 4
5 1 7 8 2
A B C
−
= = = −
4. Introduction to Matrices (cont.)
⚫ A matrix is often referred to by its size or dimensions:
m × n indicating m rows and n columns.
⚫ Matrix entries are defined first by row and then by
column.
⚫ For example, to locate the entry in matrix A defined
as aij, we look for the entry in row I, column j in
matrix A. Shown below, the entry in row 2, column 3
is a23.
11 12 13
21 22 23
31 32 33
a a a
A a a a
a a a
=
5. Introduction to Matrices (cont.)
⚫ A square matrix is a matrix with dimensions n × n,
meaning that it has the same number of rows as
columns. The 3 × 3 matrix on the previous slide is an
example of a square matrix.
⚫ A row matrix is a matrix consisting of one row with
dimensions 1 × n:
⚫ A column matrix is a matrix consisting of one column
with dimensions m × 1:
11 12 13
a a a
11
21
31
a
a
a
6. Adding and Subtracting Matrices
⚫ We use matrices to list data or to represent systems.
Because the entries are numbers, we can perform
operations on matrices. We add or subtract matrices by
adding or subtracting corresponding entries.
⚫ In order to do this, the entries must correspond.
Therefore, addition and subtraction of matrices is
only possible when the matrices have the same
dimensions.
7. Adding and Subtracting Matrices
⚫ Example: Find the sum of A and B, given
and .
a b e f
A B
c d g h
= =
8. Adding and Subtracting Matrices
⚫ Example: Find the sum of A and B, given
Add the corresponding entries.
and .
a b e f
A B
c d g h
= =
a b e f
A B
c d g h
a e b f
c g d h
+ = +
+ +
=
+ +
9. Adding and Subtracting Matrices
⚫ Example: Find the sum and difference of A and B.
2 3 8 1
and
0 1 5 4
A B
− −
= =
10. Adding and Subtracting Matrices
⚫ Example: Find the sum and difference of A and B.
2 3 8 1
and
0 1 5 4
A B
− −
= =
( ) 6 2
2 8 3 1
5 5
0 5 1 4
A B
− + + −
+ = =
+ +
11. Adding and Subtracting Matrices
⚫ Example: Find the sum and difference of A and B.
2 3 8 1
and
0 1 5 4
A B
− −
= =
( ) 6 2
2 8 3 1
5 5
0 5 1 4
A B
− + + −
+ = =
+ +
( ) 10 4
2 8 3 1
5 3
0 5 1 4
A B
−
− − − −
− = =
− −
− −
12. Finding Scalar Multiples
⚫ Besides adding and subtracting whole matrices, there
are other situations in which we need to multiply a
matrix by a constant called a scalar.
⚫ The process of scalar multiplication involves multiplying
each entry in a matrix by a scalar.
13. Finding Scalar Multiples (cont.)
⚫ Example: A university needs to add to its inventory of
computers, computer tables, and chairs in two labs due
to increased enrollment. They estimate that 15% more
equipment is needed in both labs.
Converting the data to a
matrix, we have
Lab A Lab B
Computers 15 27
Tables 16 34
Chairs 16 34 15 27
16 34
16 34
C
=
14. Finding Scalar Multiples (cont.)
⚫ To calculate how much new equipment will be needed,
we multiply all entries in matrix C by 0.15 (15%).
⚫ Because we can’t buy partial equipment, we have to
round up to the nearest integer.
( )
( ) ( )
( ) ( )
( ) ( )
0.15 15 0.15 27 2.25 4.05
0.15 0.15 16 0.15 34 2.4 5.1
0.15 16 0.15 34 2.4 5.1
C
= =
3 5 18 32
3 6 19 40 or
3 6 19 40
C
+ =
Lab A Lab B
Computers 18 32
Tables 19 40
Chairs 19 40
15. Finding Scalar Multiples (cont.)
⚫ Example:
8 1
If , what is 3 ?
5 4
A A
=
−
8 1 24 3
3 3
5 4 15 12
A
= =
− −
6 2 4 1
If , what is ?
0 3 8 2
B B
−
=
6 2 4 3 1 2
1 1
2 2 0 3 8 0 1.5 4
B
− −
= =
16. Multiplying Two Matrices
⚫ In addition to multiplying a matrix by a scalar, we can
multiply two matrices. Finding the product of two
matrices is only possible when the inner dimensions
are the same, meaning that the number of columns of
the first matrix is equal to the number of rows of the
second matrix.
⚫ If A is an m × r matrix and B is an r × n matrix, then the
product matrix AB is an m × n matrix.
⚫ If the inner dimensions do not match, the product is not
defined.
17. Multiplying Two Matrices (cont.)
⚫ To obtain the entry cij of AB, we multiply the entries in
row i in row i of A by column j in B and add.
⚫ For example, given matrices A (2 × 3) and B (3 × 3):
⚫ To obtain the entry in row 1, column 1 of AB, multiply
the first row in A by the first column of B and add:
11 12 13
11 12 13
21 22 23
21 22 23
31 32 33
and
b b b
a a a
A B b b b
a a a
b b b
= =
11
11 12 13 21 11 11 12 21 13 31
31
b
a a a b a b a b a b
b
= + +
18. Multiplying Two Matrices (cont.)
⚫ To obtain the entry in row 1, column 2 of AB, multiply
the first row of A by the second column in B, and add.
⚫ For the entry in row 1, column 3 of AB, multiply the first
row of A by the third column of B, and add.
12
11 12 13 22 11 12 12 22 13 32
32
b
a a a b a b a b a b
b
= + +
13
11 12 13 23 11 13 12 23 13 33
33
b
a a a b a b a b a b
b
= + +
19. Multiplying Two Matrices (cont.)
⚫ In the same fashion, multiply the second row of A by the
1st, 2nd, and 3rd columns of B.
⚫ Properties
⚫ Matrix multiplication is associative:
⚫ Matrix multiplication is distributive:
⚫ Matrix multiplication is not commutative:
( ) ( )
AB C A BC
=
( )
( )
C A B CA CB
A B C AC BC
+ = +
+ = +
AB BA