Operations with Matrices

Size: px
Start display at page:

Download "Operations with Matrices"

Transcription

1 Section. Equlit of Mtrices Opertions with Mtrices There re three ws to represent mtri.. A mtri cn be denoted b n uppercse letter, such s A, B, or C.. A mtri cn be denoted b representtive element enclosed in brckets, such s [ ij ], [b ij ], or [c ij ].. A mtri cn be denoted b rectngulr rr of numbers such s A = [ ij ] = m m m n n n mn Definition: Two mtrices A = [ ij ] nd B = [b ij ] re equl if the hve the sme order (m n) nd [ ij ] = [b ij ] for ll i =,,, m nd j =,,, n. In other words, if ll of the corresponding entries re equl.. but

2 Testing Mtri Equlit using Grphing Clcultor. Press [ nd ] [MATRX] [A]. Press [ nd ] [TEST] [=]. Press [ nd ] [MATRX] [B] [ENTER] If the mtrices re of the sme order, nd ll corresponding elements re equl, then the clcultor will return the vlue. CHAT Pre-Clculus Section. If the mtrices re of the sme order, nd ll corresponding elements re not ll equl, then the clcultor will return the vlue. If the mtrices do not hve the sme dimensions, ou will get nd error messge tht ss DIM MISMATCH Coping Mtri To plce the contents of mtri A into the mtri B, do the following:. Press [ nd ] [MATRX] [A]. Press [STO]. Press [ nd ] [MATRX] [B] [ENTER] A nd B re now identicl mtrices.

3 Section. Mtri Addition Definition of Mtri Addition: If A = [ ij ] nd B = [b ij ] re mtrices of order m n, their sum is the m n mtri given b A + B = [ ij + b ij ]. *The sum of two mtrices of different orders is undefined. Emple: Find the following sums. ) b) Solutions:

4 Section. Sclr Multipliction In opertions with mtrices, numbers re usull referred to s sclrs. Definition of Sclr Multipliction If A = [ ij ] is n m n mtri nd c is sclr, the sclr multiple of A b c is the m n mtri given b ca = [c ij ] Definition: The smbol A represents the dditive inverse of A nd equls (-)A. Moreover, A B = A + (-B). Emple: Consider the mtrices: A nd B

5 Section. Find the following: ) A A b) -B ) ( ) ( B B c) A B B A

6 Section. Mtri Opertions on the Grphing Clcultor Adding Mtrices If the mtrices hve the sme dimensions, the cn be dded. To dd mtri A nd mtri B, do the following:. Press [ nd ] [MATRX] [A]. Press [+]. Press [ nd ] [MATRX] [B] [ENTER] The resulting mtri is A + B. ***The sme method is used for subtrcting mtrices. Note: If the mtrices do not hve the sme dimensions, ou will get nd error messge tht ss DIM MISMATCH Sclr Multipliction To multipl sclr times the mtri A, do the following:. Enter the sclr vlue. Press [*] [ nd ] [MATRX] [A] [ENTER] Note: You cn multipl the sclr before or fter the mtri.

7 Section. Negting Mtri To chnge the signs on ll of the elements of mtri, do the following:. Enter the negtion smbol [(-)]. Press [ nd ] [MATRX] [A] [ENTER] Emple: Enter the following mtrices on our clcultor: A nd B Find the following: ) A + B b) B A B B c) -A d) A - B A A B

8 Section. Definition: The ero mtri is the m n mtri given b O = []. The ero mtri cn be n sie, nd consists entirel of eros. The ero mtri is lso the dditive identit for the set of ll m n mtrices. Properties of Mtri Addition nd Sclr Multipliction Let A, B, nd C be m n mtrices nd let c nd d be sclrs.. A + O = O + A = A. A + B = B + A. A + (B + C) = (A + B) + C. (cd)a = c(da). A = A. c(a + B) = ca + cb. (c + d)a = ca + da Note: # nd # lso men tht we cn fctor out common fctor for n mtri. The lgebr of rel numbers nd the lgebr of mtrices hve mn similrities. Emple: Solve the mtri eqution A X = B, where A nd B

9 Section. Solving for X we get X = (B A). Emple: Solve B X = A using the mtrices A nd B s given bove. A B X X

10 Section. Mtri Multipliction Definition of Mtri Multipliction: If A = [ ij ] is n m n mtri nd B = [b ij ] is n n p mtri, the product AB is n m p mtri given b AB = [c ij ] where c ij = i b i + i b i + i b i in b nj. Wht we re doing is tking the elements in the ith row of A, multipling them b the corresponding elements in the jth column of B, nd then summing these products.

11 Section. Emple: Multipl For our product, we will multipl ech row of the first mtri times ech column of the nd mtri. ( )( ) ()( ) ()( ) ( )( ) ()( ) ()( ) ( )() ()() ()() ( )() ()() ()() This entr in row, column cme from multipling row of the first mtri times column of the nd mtri. Note: In order for the product to be defined, the number of columns of the first mtri must equl the number of rows of the nd mtri. A B = AB m n n p m p Equl Dimensions of AB

12 Section. Emple: Find the following products. ) b) c) d) = undefined e)

13 Section. f) g) *Notice tht for f) nd g) tht we did not get the sme nswer. Even if AB nd BA re defined, mtri multipliction is, in generl, not commuttive. Mtri Multipliction on the Grphing Clcultor If the number of columns of the first mtri equls the number of rows of the nd mtri, the mtrices cn be multiplied. (An error messge will result otherwise.). Press [ nd ] [MATRX] [A]. Press [*]. Press [ nd ] [MATRX] [B] [ENTER] The resulting mtri is A B.

14 Section. Emple: Enter the following mtrices on our clcultor: A nd B Find AB. AB Definition: The identit mtri of order n is n n n mtri tht consists of s on its min digonl nd s everwhere else. It is denoted I n. If it is lred understood tht the mtri is squre, we cn refer to it s simpl I. emples:,, nd

15 Section. Emple: Multipl ()() ()() ()() ()() ()() ()() ()() ()() Multipling b the identit mtri gives bck the mtri we strted with. Properties of Mtri Multipliction Let A, B, nd C be mtrices nd let c be sclr.. AI n = I n A = A. A(BC) = (AB)C. A(B + C) = AB + AC. (A + B)C = AC + BC. c(ab) = (ca)b = A(cB)

16 Section. Applictions Notice how the sstem Cn be written s the mtri eqution AX = B, where A is the coefficient mtri nd X nd B re column mtrices. =

17 Section. Emple: Write the following sstem of equtions s mtri eqution AX = B. Solution: The mtri eqution is: Emple: Write the following sstem of equtions s mtri eqution AX = B. Then use Guss- Jordon elimintion on the ugmented mtri [A:B] to solve for the mtri X. Solution:

18 Section. The ugmented mtri [A:B] is: or Solving gives: The solution is X = =

19 Section. Emple: Two tennis tems submit equipment requests to their sponsors. Women s Tem Men s Tem Blls Rckets Shoes Ech cn of blls costs $, ech rcket costs $, nd ech pir of shoes costs $. Use mtri multipliction to find the totl cost of equipment for ech tem. Solution: So, the totl cost of equipment for the women s tem is $, nd the totl cost of equipment for the men s tem is $,.

Matrices and Determinants

Matrices and Determinants Nme Chpter 8 Mtrices nd Determinnts Section 8.1 Mtrices nd Systems of Equtions Objective: In this lesson you lerned how to use mtrices, Gussin elimintion, nd Guss-Jordn elimintion to solve systems of liner

More information

Introduction To Matrices MCV 4UI Assignment #1

Introduction To Matrices MCV 4UI Assignment #1 Introduction To Mtrices MCV UI Assignment # INTRODUCTION: A mtrix plurl: mtrices) is rectngulr rry of numbers rrnged in rows nd columns Exmples: ) b) c) [ ] d) Ech number ppering in the rry is sid to be

More information

Geometric Sequences. Geometric Sequence a sequence whose consecutive terms have a common ratio.

Geometric Sequences. Geometric Sequence a sequence whose consecutive terms have a common ratio. Geometric Sequences Geometric Sequence sequence whose consecutive terms hve common rtio. Geometric Sequence A sequence is geometric if the rtios of consecutive terms re the sme. 2 3 4... 2 3 The number

More information

THE DISCRIMINANT & ITS APPLICATIONS

THE DISCRIMINANT & ITS APPLICATIONS THE DISCRIMINANT & ITS APPLICATIONS The discriminnt ( Δ ) is the epression tht is locted under the squre root sign in the qudrtic formul i.e. Δ b c. For emple: Given +, Δ () ( )() The discriminnt is used

More information

INTRODUCTION TO LINEAR ALGEBRA

INTRODUCTION TO LINEAR ALGEBRA ME Applied Mthemtics for Mechnicl Engineers INTRODUCTION TO INEAR AGEBRA Mtrices nd Vectors Prof. Dr. Bülent E. Pltin Spring Sections & / ME Applied Mthemtics for Mechnicl Engineers INTRODUCTION TO INEAR

More information

Chapter 3 MATRIX. In this chapter: 3.1 MATRIX NOTATION AND TERMINOLOGY

Chapter 3 MATRIX. In this chapter: 3.1 MATRIX NOTATION AND TERMINOLOGY Chpter 3 MTRIX In this chpter: Definition nd terms Specil Mtrices Mtrix Opertion: Trnspose, Equlity, Sum, Difference, Sclr Multipliction, Mtrix Multipliction, Determinnt, Inverse ppliction of Mtrix in

More information

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics SCHOOL OF ENGINEERING & BUIL ENVIRONMEN Mthemtics An Introduction to Mtrices Definition of Mtri Size of Mtri Rows nd Columns of Mtri Mtri Addition Sclr Multipliction of Mtri Mtri Multipliction 7 rnspose

More information

MATRICES AND VECTORS SPACE

MATRICES AND VECTORS SPACE MATRICES AND VECTORS SPACE MATRICES AND MATRIX OPERATIONS SYSTEM OF LINEAR EQUATIONS DETERMINANTS VECTORS IN -SPACE AND -SPACE GENERAL VECTOR SPACES INNER PRODUCT SPACES EIGENVALUES, EIGENVECTORS LINEAR

More information

Matrices. Elementary Matrix Theory. Definition of a Matrix. Matrix Elements:

Matrices. Elementary Matrix Theory. Definition of a Matrix. Matrix Elements: Mtrices Elementry Mtrix Theory It is often desirble to use mtrix nottion to simplify complex mthemticl expressions. The simplifying mtrix nottion usully mkes the equtions much esier to hndle nd mnipulte.

More information

Algebra Of Matrices & Determinants

Algebra Of Matrices & Determinants lgebr Of Mtrices & Determinnts Importnt erms Definitions & Formule 0 Mtrix - bsic introduction: mtrix hving m rows nd n columns is clled mtrix of order m n (red s m b n mtrix) nd mtrix of order lso in

More information

MATHEMATICS AND STATISTICS 1.2

MATHEMATICS AND STATISTICS 1.2 MATHEMATICS AND STATISTICS. Apply lgebric procedures in solving problems Eternlly ssessed 4 credits Electronic technology, such s clcultors or computers, re not permitted in the ssessment of this stndr

More information

The Algebra (al-jabr) of Matrices

The Algebra (al-jabr) of Matrices Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering he lgebr (l-jbr) of Mtrices lgebr s brnch of mthemtics is much broder thn elementry lgebr ll of us studied in our high school dys. In sense

More information

Chapter 5 Determinants

Chapter 5 Determinants hpter 5 Determinnts 5. Introduction Every squre mtri hs ssocited with it sclr clled its determinnt. Given mtri, we use det() or to designte its determinnt. We cn lso designte the determinnt of mtri by

More information

Matrix Algebra. Matrix Addition, Scalar Multiplication and Transposition. Linear Algebra I 24

Matrix Algebra. Matrix Addition, Scalar Multiplication and Transposition. Linear Algebra I 24 Mtrix lger Mtrix ddition, Sclr Multipliction nd rnsposition Mtrix lger Section.. Mtrix ddition, Sclr Multipliction nd rnsposition rectngulr rry of numers is clled mtrix ( the plurl is mtrices ) nd the

More information

a a a a a a a a a a a a a a a a a a a a a a a a In this section, we introduce a general formula for computing determinants.

a a a a a a a a a a a a a a a a a a a a a a a a In this section, we introduce a general formula for computing determinants. Section 9 The Lplce Expnsion In the lst section, we defined the determinnt of (3 3) mtrix A 12 to be 22 12 21 22 2231 22 12 21. In this section, we introduce generl formul for computing determinnts. Rewriting

More information

HW3, Math 307. CSUF. Spring 2007.

HW3, Math 307. CSUF. Spring 2007. HW, Mth 7. CSUF. Spring 7. Nsser M. Abbsi Spring 7 Compiled on November 5, 8 t 8:8m public Contents Section.6, problem Section.6, problem Section.6, problem 5 Section.6, problem 7 6 5 Section.6, problem

More information

SUMMER KNOWHOW STUDY AND LEARNING CENTRE

SUMMER KNOWHOW STUDY AND LEARNING CENTRE SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18

More information

ARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac

ARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b

More information

Pre-Session Review. Part 1: Basic Algebra; Linear Functions and Graphs

Pre-Session Review. Part 1: Basic Algebra; Linear Functions and Graphs Pre-Session Review Prt 1: Bsic Algebr; Liner Functions nd Grphs A. Generl Review nd Introduction to Algebr Hierrchy of Arithmetic Opertions Opertions in ny expression re performed in the following order:

More information

MATHEMATICS FOR MANAGEMENT BBMP1103

MATHEMATICS FOR MANAGEMENT BBMP1103 T o p i c M T R I X MTHEMTICS FOR MNGEMENT BBMP Ojectives: TOPIC : MTRIX. Define mtri. ssess the clssifictions of mtrices s well s know how to perform its opertions. Clculte the determinnt for squre mtri

More information

Elements of Matrix Algebra

Elements of Matrix Algebra Elements of Mtrix Algebr Klus Neusser Kurt Schmidheiny September 30, 2015 Contents 1 Definitions 2 2 Mtrix opertions 3 3 Rnk of Mtrix 5 4 Specil Functions of Qudrtic Mtrices 6 4.1 Trce of Mtrix.........................

More information

September 13 Homework Solutions

September 13 Homework Solutions College of Engineering nd Computer Science Mechnicl Engineering Deprtment Mechnicl Engineering 5A Seminr in Engineering Anlysis Fll Ticket: 5966 Instructor: Lrry Cretto Septemer Homework Solutions. Are

More information

4 7x =250; 5 3x =500; Read section 3.3, 3.4 Announcements: Bell Ringer: Use your calculator to solve

4 7x =250; 5 3x =500; Read section 3.3, 3.4 Announcements: Bell Ringer: Use your calculator to solve Dte: 3/14/13 Objective: SWBAT pply properties of exponentil functions nd will pply properties of rithms. Bell Ringer: Use your clcultor to solve 4 7x =250; 5 3x =500; HW Requests: Properties of Log Equtions

More information

Lecture 3. In this lecture, we will discuss algorithms for solving systems of linear equations.

Lecture 3. In this lecture, we will discuss algorithms for solving systems of linear equations. Lecture 3 3 Solving liner equtions In this lecture we will discuss lgorithms for solving systems of liner equtions Multiplictive identity Let us restrict ourselves to considering squre mtrices since one

More information

Logarithms. Logarithm is another word for an index or power. POWER. 2 is the power to which the base 10 must be raised to give 100.

Logarithms. Logarithm is another word for an index or power. POWER. 2 is the power to which the base 10 must be raised to give 100. Logrithms. Logrithm is nother word for n inde or power. THIS IS A POWER STATEMENT BASE POWER FOR EXAMPLE : We lred know tht; = NUMBER 10² = 100 This is the POWER Sttement OR 2 is the power to which the

More information

Reference : Croft & Davison, Chapter 12, Blocks 1,2. A matrix ti is a rectangular array or block of numbers usually enclosed in brackets.

Reference : Croft & Davison, Chapter 12, Blocks 1,2. A matrix ti is a rectangular array or block of numbers usually enclosed in brackets. I MATRIX ALGEBRA INTRODUCTION TO MATRICES Referene : Croft & Dvison, Chpter, Blos, A mtri ti is retngulr rr or lo of numers usull enlosed in rets. A m n mtri hs m rows nd n olumns. Mtri Alger Pge If the

More information

DETERMINANTS. All Mathematical truths are relative and conditional. C.P. STEINMETZ

DETERMINANTS. All Mathematical truths are relative and conditional. C.P. STEINMETZ All Mthemticl truths re reltive nd conditionl. C.P. STEINMETZ 4. Introduction DETERMINANTS In the previous chpter, we hve studied bout mtrices nd lgebr of mtrices. We hve lso lernt tht system of lgebric

More information

Read section 3.3, 3.4 Announcements:

Read section 3.3, 3.4 Announcements: Dte: 3/1/13 Objective: SWBAT pply properties of exponentil functions nd will pply properties of rithms. Bell Ringer: 1. f x = 3x 6, find the inverse, f 1 x., Using your grphing clcultor, Grph 1. f x,f

More information

ECON 331 Lecture Notes: Ch 4 and Ch 5

ECON 331 Lecture Notes: Ch 4 and Ch 5 Mtrix Algebr ECON 33 Lecture Notes: Ch 4 nd Ch 5. Gives us shorthnd wy of writing lrge system of equtions.. Allows us to test for the existnce of solutions to simultneous systems. 3. Allows us to solve

More information

Chapter 1: Fundamentals

Chapter 1: Fundamentals Chpter 1: Fundmentls 1.1 Rel Numbers Types of Rel Numbers: Nturl Numbers: {1, 2, 3,...}; These re the counting numbers. Integers: {... 3, 2, 1, 0, 1, 2, 3,...}; These re ll the nturl numbers, their negtives,

More information

Engineering Analysis ENG 3420 Fall Dan C. Marinescu Office: HEC 439 B Office hours: Tu-Th 11:00-12:00

Engineering Analysis ENG 3420 Fall Dan C. Marinescu Office: HEC 439 B Office hours: Tu-Th 11:00-12:00 Engineering Anlysis ENG 3420 Fll 2009 Dn C. Mrinescu Office: HEC 439 B Office hours: Tu-Th 11:00-12:00 Lecture 13 Lst time: Problem solving in preprtion for the quiz Liner Algebr Concepts Vector Spces,

More information

How do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?

How do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique? XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk bout solving systems of liner equtions. These re problems tht give couple of equtions with couple of unknowns, like: 6 2 3 7 4

More information

CSCI 5525 Machine Learning

CSCI 5525 Machine Learning CSCI 555 Mchine Lerning Some Deini*ons Qudrtic Form : nn squre mtri R n n : n vector R n the qudrtic orm: It is sclr vlue. We oten implicitly ssume tht is symmetric since / / I we write it s the elements

More information

Matrix Solution to Linear Equations and Markov Chains

Matrix Solution to Linear Equations and Markov Chains Trding Systems nd Methods, Fifth Edition By Perry J. Kufmn Copyright 2005, 2013 by Perry J. Kufmn APPENDIX 2 Mtrix Solution to Liner Equtions nd Mrkov Chins DIRECT SOLUTION AND CONVERGENCE METHOD Before

More information

Consolidation Worksheet

Consolidation Worksheet Cmbridge Essentils Mthemtics Core 8 NConsolidtion Worksheet N Consolidtion Worksheet Work these out. 8 b 7 + 0 c 6 + 7 5 Use the number line to help. 2 Remember + 2 2 +2 2 2 + 2 Adding negtive number is

More information

Quotient Rule: am a n = am n (a 0) Negative Exponents: a n = 1 (a 0) an Power Rules: (a m ) n = a m n (ab) m = a m b m

Quotient Rule: am a n = am n (a 0) Negative Exponents: a n = 1 (a 0) an Power Rules: (a m ) n = a m n (ab) m = a m b m Formuls nd Concepts MAT 099: Intermedite Algebr repring for Tests: The formuls nd concepts here m not be inclusive. You should first tke our prctice test with no notes or help to see wht mteril ou re comfortble

More information

MathCity.org Merging man and maths

MathCity.org Merging man and maths MthCity.org Merging mn nd mths Exercise.8 (s) Pge 46 Textbook of Algebr nd Trigonometry for Clss XI Avilble online @ http://, Version: 3.0 Question # Opertion performed on the two-member set G = {0, is

More information

Operations with Polynomials

Operations with Polynomials 38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: How to identify the leding coefficients nd degrees of polynomils How to dd nd subtrct polynomils How to multiply polynomils

More information

DETERMINANTS. All Mathematical truths are relative and conditional. C.P. STEINMETZ

DETERMINANTS. All Mathematical truths are relative and conditional. C.P. STEINMETZ DETERMINANTS Chpter 4 All Mthemticl truths re reltive nd conditionl. C.P. STEINMETZ 4. Introduction In the previous chpter, we hve studied bout mtrices nd lgebr of mtrices. We hve lso lernt tht sstem of

More information

5.2 Exponent Properties Involving Quotients

5.2 Exponent Properties Involving Quotients 5. Eponent Properties Involving Quotients Lerning Objectives Use the quotient of powers property. Use the power of quotient property. Simplify epressions involving quotient properties of eponents. Use

More information

Chapter 1: Logarithmic functions and indices

Chapter 1: Logarithmic functions and indices Chpter : Logrithmic functions nd indices. You cn simplify epressions y using rules of indices m n m n m n m n ( m ) n mn m m m m n m m n Emple Simplify these epressions: 5 r r c 4 4 d 6 5 e ( ) f ( ) 4

More information

1 Linear Least Squares

1 Linear Least Squares Lest Squres Pge 1 1 Liner Lest Squres I will try to be consistent in nottion, with n being the number of dt points, nd m < n being the number of prmeters in model function. We re interested in solving

More information

Matrix- System of rows and columns each position in a matrix has a purpose. 5 Ex: 5. Ex:

Matrix- System of rows and columns each position in a matrix has a purpose. 5 Ex: 5. Ex: Mtries Prelulus Mtri- Sstem of rows n olumns eh position in mtri hs purpose. Element- Eh vlue in the mtri mens the element in the n row, r olumn Dimensions- How mn rows b number of olumns Ientif the element:

More information

Here we study square linear systems and properties of their coefficient matrices as they relate to the solution set of the linear system.

Here we study square linear systems and properties of their coefficient matrices as they relate to the solution set of the linear system. Section 24 Nonsingulr Liner Systems Here we study squre liner systems nd properties of their coefficient mtrices s they relte to the solution set of the liner system Let A be n n Then we know from previous

More information

fractions Let s Learn to

fractions Let s Learn to 5 simple lgebric frctions corne lens pupil retin Norml vision light focused on the retin concve lens Shortsightedness (myopi) light focused in front of the retin Corrected myopi light focused on the retin

More information

Chapter 2. Vectors. 2.1 Vectors Scalars and Vectors

Chapter 2. Vectors. 2.1 Vectors Scalars and Vectors Chpter 2 Vectors 2.1 Vectors 2.1.1 Sclrs nd Vectors A vector is quntity hving both mgnitude nd direction. Emples of vector quntities re velocity, force nd position. One cn represent vector in n-dimensionl

More information

AQA Further Pure 1. Complex Numbers. Section 1: Introduction to Complex Numbers. The number system

AQA Further Pure 1. Complex Numbers. Section 1: Introduction to Complex Numbers. The number system Complex Numbers Section 1: Introduction to Complex Numbers Notes nd Exmples These notes contin subsections on The number system Adding nd subtrcting complex numbers Multiplying complex numbers Complex

More information

Matrix Eigenvalues and Eigenvectors September 13, 2017

Matrix Eigenvalues and Eigenvectors September 13, 2017 Mtri Eigenvlues nd Eigenvectors September, 7 Mtri Eigenvlues nd Eigenvectors Lrry Cretto Mechnicl Engineering 5A Seminr in Engineering Anlysis September, 7 Outline Review lst lecture Definition of eigenvlues

More information

Matrix & Vector Basic Linear Algebra & Calculus

Matrix & Vector Basic Linear Algebra & Calculus Mtrix & Vector Bsic Liner lgebr & lculus Wht is mtrix? rectngulr rry of numbers (we will concentrte on rel numbers). nxm mtrix hs n rows n m columns M x4 M M M M M M M M M M M M 4 4 4 First row Secon row

More information

MATRIX DEFINITION A matrix is any doubly subscripted array of elements arranged in rows and columns.

MATRIX DEFINITION A matrix is any doubly subscripted array of elements arranged in rows and columns. 4.5 THEORETICL SOIL MECHNICS Vector nd Mtrix lger Review MTRIX DEFINITION mtrix is ny douly suscripted rry of elements rrnged in rows nd columns. m - Column Revised /0 n -Row m,,,,,, n n mn ij nd Order

More information

8 factors of x. For our second example, let s raise a power to a power:

8 factors of x. For our second example, let s raise a power to a power: CH 5 THE FIVE LAWS OF EXPONENTS EXPONENTS WITH VARIABLES It s no time for chnge in tctics, in order to give us deeper understnding of eponents. For ech of the folloing five emples, e ill stretch nd squish,

More information

Multivariate problems and matrix algebra

Multivariate problems and matrix algebra University of Ferrr Stefno Bonnini Multivrite problems nd mtrix lgebr Multivrite problems Multivrite sttisticl nlysis dels with dt contining observtions on two or more chrcteristics (vribles) ech mesured

More information

Identify graphs of linear inequalities on a number line.

Identify graphs of linear inequalities on a number line. COMPETENCY 1.0 KNOWLEDGE OF ALGEBRA SKILL 1.1 Identify grphs of liner inequlities on number line. - When grphing first-degree eqution, solve for the vrible. The grph of this solution will be single point

More information

Year 11 Matrices. A row of seats goes across an auditorium So Rows are horizontal. The columns of the Parthenon stand upright and Columns are vertical

Year 11 Matrices. A row of seats goes across an auditorium So Rows are horizontal. The columns of the Parthenon stand upright and Columns are vertical Yer 11 Mtrices Terminology: A single MATRIX (singulr) or Mny MATRICES (plurl) Chpter 3A Intro to Mtrices A mtrix is escribe s n orgnise rry of t. We escribe the ORDER of Mtrix (it's size) by noting how

More information

ES.182A Topic 30 Notes Jeremy Orloff

ES.182A Topic 30 Notes Jeremy Orloff ES82A opic 3 Notes Jerem Orloff 3 Non-independent vribles: chin rule Recll the chin rule: If w = f, ; nd = r, t, = r, t then = + r t r t r t = + t t t r nfortuntel, sometimes there re more complicted reltions

More information

Introduction to Algebra - Part 2

Introduction to Algebra - Part 2 Alger Module A Introduction to Alger - Prt Copright This puliction The Northern Alert Institute of Technolog 00. All Rights Reserved. LAST REVISED Oct., 008 Introduction to Alger - Prt Sttement of Prerequisite

More information

Chapter 2. Determinants

Chapter 2. Determinants Chpter Determinnts The Determinnt Function Recll tht the X mtrix A c b d is invertible if d-bc0. The expression d-bc occurs so frequently tht it hs nme; it is clled the determinnt of the mtrix A nd is

More information

Equations, expressions and formulae

Equations, expressions and formulae Get strted 2 Equtions, epressions nd formule This unit will help you to work with equtions, epressions nd formule. AO1 Fluency check 1 Work out 2 b 2 c 7 2 d 7 2 2 Simplify by collecting like terms. b

More information

On the free product of ordered groups

On the free product of ordered groups rxiv:703.0578v [mth.gr] 6 Mr 207 On the free product of ordered groups A. A. Vinogrdov One of the fundmentl questions of the theory of ordered groups is wht bstrct groups re orderble. E. P. Shimbirev [2]

More information

A-Level Mathematics Transition Task (compulsory for all maths students and all further maths student)

A-Level Mathematics Transition Task (compulsory for all maths students and all further maths student) A-Level Mthemtics Trnsition Tsk (compulsory for ll mths students nd ll further mths student) Due: st Lesson of the yer. Length: - hours work (depending on prior knowledge) This trnsition tsk provides revision

More information

Loudoun Valley High School Calculus Summertime Fun Packet

Loudoun Valley High School Calculus Summertime Fun Packet Loudoun Vlley High School Clculus Summertime Fun Pcket We HIGHLY recommend tht you go through this pcket nd mke sure tht you know how to do everything in it. Prctice the problems tht you do NOT remember!

More information

Lecture Solution of a System of Linear Equation

Lecture Solution of a System of Linear Equation ChE Lecture Notes, Dept. of Chemicl Engineering, Univ. of TN, Knoville - D. Keffer, 5/9/98 (updted /) Lecture 8- - Solution of System of Liner Eqution 8. Why is it importnt to e le to solve system of liner

More information

M344 - ADVANCED ENGINEERING MATHEMATICS

M344 - ADVANCED ENGINEERING MATHEMATICS M3 - ADVANCED ENGINEERING MATHEMATICS Lecture 18: Lplce s Eqution, Anltic nd Numericl Solution Our emple of n elliptic prtil differentil eqution is Lplce s eqution, lso clled the Diffusion Eqution. If

More information

Homework 5 solutions

Homework 5 solutions Section.: E ; AP, Section.: E,,; AP,,, Section. Homework solutions. Consider the two upper-tringulr mtrices: b b b A, B b b. b () Show tht their product C = AB is lso upper-tringulr. The product is b b

More information

1. Extend QR downwards to meet the x-axis at U(6, 0). y

1. Extend QR downwards to meet the x-axis at U(6, 0). y In the digrm, two stright lines re to be drwn through so tht the lines divide the figure OPQRST into pieces of equl re Find the sum of the slopes of the lines R(6, ) S(, ) T(, 0) Determine ll liner functions

More information

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT SCHOOL OF ENGINEERING & BUIL ENVIRONMEN MARICES FOR ENGINEERING Dr Clum Mcdonld Contents Introduction Definitions Wht is mtri? Rows nd columns of mtri Order of mtri Element of mtri Equlity of mtrices Opertions

More information

USA Mathematical Talent Search Round 1 Solutions Year 21 Academic Year

USA Mathematical Talent Search Round 1 Solutions Year 21 Academic Year 1/1/21. Fill in the circles in the picture t right with the digits 1-8, one digit in ech circle with no digit repeted, so tht no two circles tht re connected by line segment contin consecutive digits.

More information

Math 4310 Solutions to homework 1 Due 9/1/16

Math 4310 Solutions to homework 1 Due 9/1/16 Mth 4310 Solutions to homework 1 Due 9/1/16 1. Use the Eucliden lgorithm to find the following gretest common divisors. () gcd(252, 180) = 36 (b) gcd(513, 187) = 1 (c) gcd(7684, 4148) = 68 252 = 180 1

More information

308K. 1 Section 3.2. Zelaya Eufemia. 1. Example 1: Multiplication of Matrices: X Y Z R S R S X Y Z. By associativity we have to choices:

308K. 1 Section 3.2. Zelaya Eufemia. 1. Example 1: Multiplication of Matrices: X Y Z R S R S X Y Z. By associativity we have to choices: 8K Zely Eufemi Section 2 Exmple : Multipliction of Mtrices: X Y Z T c e d f 2 R S X Y Z 2 c e d f 2 R S 2 By ssocitivity we hve to choices: OR: X Y Z R S cr ds er fs X cy ez X dy fz 2 R S 2 Suggestion

More information

Matrices 13: determinant properties and rules continued

Matrices 13: determinant properties and rules continued Mtrices : determinnt properties nd rules continued nthony Rossiter http://controleduction.group.shef.c.uk/indexwebbook.html http://www.shef.c.uk/cse Deprtment of utomtic Control nd Systems Engineering

More information

Equations and Inequalities

Equations and Inequalities Equtions nd Inequlities Equtions nd Inequlities Curriculum Redy ACMNA: 4, 5, 6, 7, 40 www.mthletics.com Equtions EQUATIONS & Inequlities & INEQUALITIES Sometimes just writing vribles or pronumerls in

More information

7h1 Simplifying Rational Expressions. Goals:

7h1 Simplifying Rational Expressions. Goals: h Simplifying Rtionl Epressions Gols Fctoring epressions (common fctor, & -, no fctoring qudrtics) Stting restrictions Epnding rtionl epressions Simplifying (reducin rtionl epressions (Kürzen) Adding nd

More information

Calculus 2: Integration. Differentiation. Integration

Calculus 2: Integration. Differentiation. Integration Clculus 2: Integrtion The reverse process to differentition is known s integrtion. Differentition f() f () Integrtion As it is the opposite of finding the derivtive, the function obtined b integrtion is

More information

Lesson 1: Quadratic Equations

Lesson 1: Quadratic Equations Lesson 1: Qudrtic Equtions Qudrtic Eqution: The qudrtic eqution in form is. In this section, we will review 4 methods of qudrtic equtions, nd when it is most to use ech method. 1. 3.. 4. Method 1: Fctoring

More information

Things to Memorize: A Partial List. January 27, 2017

Things to Memorize: A Partial List. January 27, 2017 Things to Memorize: A Prtil List Jnury 27, 2017 Chpter 2 Vectors - Bsic Fcts A vector hs mgnitude (lso clled size/length/norm) nd direction. It does not hve fixed position, so the sme vector cn e moved

More information

Chapter 14. Matrix Representations of Linear Transformations

Chapter 14. Matrix Representations of Linear Transformations Chpter 4 Mtrix Representtions of Liner Trnsformtions When considering the Het Stte Evolution, we found tht we could describe this process using multipliction by mtrix. This ws nice becuse computers cn

More information

MTH 4-16a Trigonometry

MTH 4-16a Trigonometry MTH 4-16 Trigonometry Level 4 [UNIT 5 REVISION SECTION ] I cn identify the opposite, djcent nd hypotenuse sides on right-ngled tringle. Identify the opposite, djcent nd hypotenuse in the following right-ngled

More information

Boolean Algebra. Boolean Algebras

Boolean Algebra. Boolean Algebras Boolen Algebr Boolen Algebrs A Boolen lgebr is set B of vlues together with: - two binry opertions, commonly denoted by + nd, - unry opertion, usully denoted by or ~ or, - two elements usully clled zero

More information

50. Use symmetry to evaluate xx D is the region bounded by the square with vertices 5, Prove Property 11. y y CAS

50. Use symmetry to evaluate xx D is the region bounded by the square with vertices 5, Prove Property 11. y y CAS 68 CHAPTE MULTIPLE INTEGALS 46. e da, 49. Evlute tn 3 4 da, where,. [Hint: Eploit the fct tht is the disk with center the origin nd rdius is smmetric with respect to both es.] 5. Use smmetr to evlute 3

More information

4.5 JACOBI ITERATION FOR FINDING EIGENVALUES OF A REAL SYMMETRIC MATRIX. be a real symmetric matrix. ; (where we choose θ π for.

4.5 JACOBI ITERATION FOR FINDING EIGENVALUES OF A REAL SYMMETRIC MATRIX. be a real symmetric matrix. ; (where we choose θ π for. 4.5 JACOBI ITERATION FOR FINDING EIGENVALUES OF A REAL SYMMETRIC MATRIX Some reliminries: Let A be rel symmetric mtrix. Let Cos θ ; (where we choose θ π for Cos θ 4 purposes of convergence of the scheme)

More information

Bridging the gap: GCSE AS Level

Bridging the gap: GCSE AS Level Bridging the gp: GCSE AS Level CONTENTS Chpter Removing rckets pge Chpter Liner equtions Chpter Simultneous equtions 8 Chpter Fctors 0 Chpter Chnge the suject of the formul Chpter 6 Solving qudrtic equtions

More information

Sample pages. 9:04 Equations with grouping symbols

Sample pages. 9:04 Equations with grouping symbols Equtions 9 Contents I know the nswer is here somewhere! 9:01 Inverse opertions 9:0 Solving equtions Fun spot 9:0 Why did the tooth get dressed up? 9:0 Equtions with pronumerls on both sides GeoGebr ctivity

More information

PART 1 MULTIPLE CHOICE Circle the appropriate response to each of the questions below. Each question has a value of 1 point.

PART 1 MULTIPLE CHOICE Circle the appropriate response to each of the questions below. Each question has a value of 1 point. PART MULTIPLE CHOICE Circle the pproprite response to ech of the questions below. Ech question hs vlue of point.. If in sequence the second level difference is constnt, thn the sequence is:. rithmetic

More information

Reference. Vector Analysis Chapter 2

Reference. Vector Analysis Chapter 2 Reference Vector nlsis Chpter Sttic Electric Fields (3 Weeks) Chpter 3.3 Coulomb s Lw Chpter 3.4 Guss s Lw nd pplictions Chpter 3.5 Electric Potentil Chpter 3.6 Mteril Medi in Sttic Electric Field Chpter

More information

The Trapezoidal Rule

The Trapezoidal Rule _.qd // : PM Pge 9 SECTION. Numericl Integrtion 9 f Section. The re of the region cn e pproimted using four trpezoids. Figure. = f( ) f( ) n The re of the first trpezoid is f f n. Figure. = Numericl Integrtion

More information

Linear Inequalities. Work Sheet 1

Linear Inequalities. Work Sheet 1 Work Sheet 1 Liner Inequlities Rent--Hep, cr rentl compny,chrges $ 15 per week plus $ 0.0 per mile to rent one of their crs. Suppose you re limited y how much money you cn spend for the week : You cn spend

More information

HQPD - ALGEBRA I TEST Record your answers on the answer sheet.

HQPD - ALGEBRA I TEST Record your answers on the answer sheet. HQPD - ALGEBRA I TEST Record your nswers on the nswer sheet. Choose the best nswer for ech. 1. If 7(2d ) = 5, then 14d 21 = 5 is justified by which property? A. ssocitive property B. commuttive property

More information

Fundamentals of Linear Algebra

Fundamentals of Linear Algebra -7/8-797 Mchine Lerning for Signl rocessing Fundmentls of Liner Alger Administrivi Registrtion: Anone on witlist still? Homework : Will e hnded out with clss Liner lger Clss - Sep Instructor: Bhiksh Rj

More information

12.1 Introduction to Rational Expressions

12.1 Introduction to Rational Expressions . Introduction to Rtionl Epressions A rtionl epression is rtio of polynomils; tht is, frction tht hs polynomil s numertor nd/or denomintor. Smple rtionl epressions: 0 EVALUATING RATIONAL EXPRESSIONS To

More information

Add and Subtract Rational Expressions. You multiplied and divided rational expressions. You will add and subtract rational expressions.

Add and Subtract Rational Expressions. You multiplied and divided rational expressions. You will add and subtract rational expressions. TEKS 8. A..A, A.0.F Add nd Subtrct Rtionl Epressions Before Now You multiplied nd divided rtionl epressions. You will dd nd subtrct rtionl epressions. Why? So you cn determine monthly cr lon pyments, s

More information

In this skill we review equations that involve percents. review the meaning of proportion.

In this skill we review equations that involve percents. review the meaning of proportion. 6 MODULE 5. PERCENTS 5b Solving Equtions Mening of Proportion In this skill we review equtions tht involve percents. review the mening of proportion. Our first tsk is to Proportions. A proportion is sttement

More information

Lesson 4 Linear Algebra

Lesson 4 Linear Algebra Lesso Lier Algebr A fmily of vectors is lierly idepedet if oe of them c be writte s lier combitio of fiitely my other vectors i the collectio. Cosider m lierly idepedet equtios i ukows:, +, +... +, +,

More information

Module 6: LINEAR TRANSFORMATIONS

Module 6: LINEAR TRANSFORMATIONS Module 6: LINEAR TRANSFORMATIONS. Trnsformtions nd mtrices Trnsformtions re generliztions of functions. A vector x in some set S n is mpped into m nother vector y T( x). A trnsformtion is liner if, for

More information

Lecture 8 Wrap-up Part1, Matlab

Lecture 8 Wrap-up Part1, Matlab Lecture 8 Wrp-up Prt1, Mtlb Homework Polic Plese stple our homework (ou will lose 10% credit if not stpled or secured) Submit ll problems in order. This mens to plce ever item relting to problem 3 (our

More information

10.2 The Ellipse and the Hyperbola

10.2 The Ellipse and the Hyperbola CHAPTER 0 Conic Sections Solve. 97. Two surveors need to find the distnce cross lke. The plce reference pole t point A in the digrm. Point B is meters est nd meter north of the reference point A. Point

More information

On the diagram below the displacement is represented by the directed line segment OA.

On the diagram below the displacement is represented by the directed line segment OA. Vectors Sclrs nd Vectors A vector is quntity tht hs mgnitude nd direction. One exmple of vector is velocity. The velocity of n oject is determined y the mgnitude(speed) nd direction of trvel. Other exmples

More information

Is there an easy way to find examples of such triples? Why yes! Just look at an ordinary multiplication table to find them!

Is there an easy way to find examples of such triples? Why yes! Just look at an ordinary multiplication table to find them! PUSHING PYTHAGORAS 009 Jmes Tnton A triple of integers ( bc,, ) is clled Pythgoren triple if exmple, some clssic triples re ( 3,4,5 ), ( 5,1,13 ), ( ) fond of ( 0,1,9 ) nd ( 119,10,169 ). + b = c. For

More information

How do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?

How do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique? XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk out solving systems of liner equtions. These re prolems tht give couple of equtions with couple of unknowns, like: 6= x + x 7=

More information

STRAND B: NUMBER THEORY

STRAND B: NUMBER THEORY Mthemtics SKE, Strnd B UNIT B Indices nd Fctors: Tet STRAND B: NUMBER THEORY B Indices nd Fctors Tet Contents Section B. Squres, Cubes, Squre Roots nd Cube Roots B. Inde Nottion B. Fctors B. Prime Fctors,

More information

The Islamic University of Gaza Faculty of Engineering Civil Engineering Department. Numerical Analysis ECIV Chapter 11

The Islamic University of Gaza Faculty of Engineering Civil Engineering Department. Numerical Analysis ECIV Chapter 11 The Islmic University of Gz Fculty of Engineering Civil Engineering Deprtment Numericl Anlysis ECIV 6 Chpter Specil Mtrices nd Guss-Siedel Associte Prof Mzen Abultyef Civil Engineering Deprtment, The Islmic

More information