Example 1.1 Use an augmented matrix to mimic the elimination method for solving the following linear system of equations.

Transcription

1 MTH 261 Mr Simods class Example 11 Use a augmeted matrix to mimic the elimiatio method for solvig the followig liear system of equatios 2x1 3x2 8 6x1 x2 36 Example 12 Use the method of Gaussia elimiatio to fid a echelo form of the augmeted matrix represetatio for each of the followig systems of equatios ad use that matrix to determie the solutio to the system of equatios a 2x1 5x2 3x3 23 5x1 x2 2x3 7 x1 3x2 x3 3 b x1 2x2 x4 4 2x1 3x2 2x3 2x4 3 x2 2x3 11 5x1 10x2 3x4 1 Example 13 Several augmeted row echelo form matrices are give below (ad o the ext page) For each matrix, idetify the pivot colums ad state the ature of the solutio set for the associated system of equatios a A b B c C Example 14 Solve the followig systems of equatios usig the Gauss Jorda elimiatio method a 2x1 3x2 2 6x1 6x2 1 b 2x2 6x3 2 4x1 x2 3x3 1 x1 3x2 8x3 4 c x1 6x2 2x3 9 3x1 2x2 x3 5x4 1 2x1 4x2 x3 5x4 8 3x1 x2 x3 7x4 6 Systems, Matrices, Gauss, ad Jorda Sectios 11, 12 ad 16 11

2 MTH 261 Mr Simods class Example 15 State a geeral solutio to the followig systems of equatios as well as two specific solutios to the system a x1 2x2 2x3 3x4 2 x2 x3 2x4 3 b 2x1 6x2 5x3 2 x1 3x2 3x3 1 Example 16 Uder what coditios is the followig system of equatios cosistet? x1 x2 2 x3 a x1 3 x2 x3 b 2x1 8x2 5x3 c Example 17: Applicatio: Network Aalysis A etwork is most easily thought of as a city street system The itersectios are techically called odes or juctios ad each directed stretch of road betwee itersectios is called a brach Because braches are directed, if there is a two way street betwee two itersectios the correspodig etwork will have two braches betwee the correspodig odes We assig values or variables to each brach; those values ad variables could coceptually represet flow rates or flow amouts alog those braches I order for the etwork to be valid, the total flow ito the etwork must equal the total flow out of the etwork The values ad variables i Figure 1 represet traffic flow rates (vehicles/quarter hour) i a small sectio of a city street system Let s determie the miimum ad maximum flow rates through each of the variable braches x 1 50 A B x 4 x x 3 25 D C Figure 1: Network Diagram S ystems, Matrices, Gauss, ad Jorda Sectios 11, 12 ad 16

3 MTH 261 Mr Simods class Defiitios The Gaussia elimiatio process for solvig systems of liear equatios is predicated upo the three elemetary row operatios The three elemetary row operatios are: 1 Iterchagig two rows of the matrix 2 Replacig oe row of the matrix with a o zero multiple of itself 3 Replacig oe row of a matrix with itself added to a multiple of aother row of the matrix Two matrices are said to be row equivalet if oe matrix ca be trasformed ito the other via a series of elemetary row operatios The first o zero etry of ay row of a matrix is called the leadig etry of that row A matrix is said to be i (row) echelo form if the matrix satisfies both of the followig properties i Every row that cotais othig but zeros occurs at the bottom of the matrix ii The leadig etry of ay o zero row appears to the right of the leadig etry i the row directly above it The process of trasformig a matrix ito a row equivalet matrix of echelo form is called the Gaussia elimiatio process Defiitios ad Theorem 11 A pivot positio i a matrix is a etry positio that correspods to a leadig etry i a row echelo form of the matrix Ay colum that cotais a pivot etry is called a pivot colum If A is a augmeted matrix represetig a system of equatio with variables, the: the system of equatios has exactly oe solutio if ad oly if A has exactly pivot colums all of which lie to the left of the augmet lie; the system of equatios has o solutios if ad oly if the right most colum of A is a pivot colum; the system of equatios has a ulimited umber of solutios if ad oly if there are less tha pivot colums i A ad the right most colum of A is ot a pivot colum A system of equatios with at least oe solutio is called cosistet A system of equatios with o solutios is called icosistet Systems, Matrices, Gauss, ad Jorda Sectios 11, 12 ad 16 13

4 MTH 261 Mr Simods class Gauss-Jorda Elimiatio Method The Gauss Jorda elimiatio method follows the same row maipulatio rules as Gaussia elimiatio What distiguishes the two methods is that the Gaussia method stops oce you ve reduced the matrix to row echelo form where as the Gauss Jordaia method requires you to further reduce the matrix to reduced (row) echelo form That is, the simplified matrix must meet each of the followig properties: i Every row that cotais othig but zeros occurs at the bottom of the matrix ii The leadig etry of ay o zero row appears to the right of the leadig etry i the row directly above it iii All etries directly above or below a leadig etry are zero iv Every leadig etry is 1 Whe maipulatig a matrix ito reduced row echelo form, there are essetially three tasks that eed to be completed I the list below, you always wat to complete task A first Tasks B ad C ca be doe i either order A Maipulate every etry below a leadig etry to zero This task should be worked left toright; ie, first create the ecessary zeros i the first colum, the the secod colum, the the third colum, etc Remember that the leadig etries eed to move rightward as you move dow the rows of the matrix Occasioally you will eed to swap rows to maitai this cascadig effect Always use the row cotaiig the leadig etry to create the zeros below that leadig etry B Maipulate every etry above a leadig etry to zero This task should be worked rightto left Always use the row cotaiig the leadig etry to create the zeros above that leadig etry C Multiply each of the rows by the ecessary costats so that each leadig etry is 1 Defiitios Suppose that A is the augmeted represetatio of a cosistet system with variables that has a ulimited umber of solutios The free variables i the geeral solutio set correspod to the o pivot colums of A that lie to the left of the augmet lie The other variables i the system are called basic variables The value of ay give basic variable might be fixed or it might be depedet upo the value(s) assiged to oe or more of the free variables Specific solutios to the system are determied by assigig specific values to the free variables ad the determiig the associated values of the basic variables 14 S ystems, Matrices, Gauss, ad Jorda Sectios 11, 12 ad 16

5 MTH 261 Mr Simods class Example Simplify Figure 1 ad illustrate the process o Example Let a1 4, a2 1, ad b 13 as a liear combiatio of a 1 ad a 2 Express b Example Let A ad 3 b h 2 Fid the value of h Figure 1: if b is i the spa of the colums of A Example Let A 0 5 2, B , u 2, ad 2 7 products (where possible): Au, Av, Bu, ad B v 2 v 4 8 Fid each of the followig Example 25 Write 1 2 5x x x 3 3 as a vector equatio Example 26 Write the system 2x1 3x2 x3 2x4 0 5x1 x2 x4 9 as a matrix equatio of form Axb Vector Equatios Sectio 13/Matrix Equatios Sectio 14 21

6 MTH 261 Mr Simods class Example 27: Applicatio: Balacig Chemical Equatios Ethae ad Oxyge combie to produce Carbo Dioxide ad steam Formally, this is represeted by the equatio C2 H6 O2 CO2 H2 O Let s put our ewfoud skills to use ad balace this equatio Speakig of puttig thigs to use let s use our calculator to fid the RREF form of the matrix Defiitio 27: Applicatio: Ceter of Mass The ceter of mass (or ceter of gravity) of a object is the average of the product of mass poits i the object with their relative distace from a fixed referece poit The cocept is most easily uderstood if o thiks about a teeter totter Whe two people of differet mass sit o a teeter totter of uiform desity ad thickess, the ceter of mass is clearly goig to be closer to the heavier perso tha the lighter perso; this pheomeo is reflected i the fact that if the teeter totter is to stay if balace, the heavier perso eeds to sit closer to the tippig poit tha the lighter perso Ufortuately, we ca t defie the ceter of mass i all cases as the balace poit, because the ceter of mass is frequetly ot eve o the object! This is easy to see if you thik about a washer (as i uts ad bolts) of uiform desity; the ceter of mass is clearly the ceter poit of the hole i the middle of the washer For most objects, it takes a double or triple itegral to calculate the ceter of mass I some simple situatios, however, the poit ca be determied by a simple formula For example, if we have a triagular lamia of uiform mass desity refereced to the xy plae, the ceter of mass ( v ) ca be the origi to the three vertices of the triagle; for ease of referece, we shall cotextually refer to these vectors as poits determied usig the formula v v v v where v 1, v 2, ad v 3 represet the vectors from That formula is a special case of the more geeral formula for the ceter of mass of several mass poits Example 28 Fid the ceter of mass of the triagular lamia outlied i Figure 2 assumig that the lamia has uiform desity ad thickess Example 29 I Example 28 we foud the ceter of mass usig the formula v m v ad assumig that there was 1 g of mass at 1 i i m i 1 each of the vertices Suppose that we had 9 additio grams of mass to distribute amog the vertices How should the mass be distributed so that the ceter of mass of the lamia shifts to the poit 3, 2? Figure 2: lamia of uiform desity ad thickess 22 V ector Equatios Sectio 13/Matrix Equatios Sectio 14

7 MTH 261 Mr Simods class Defiitios A colum vector is a matrix with oly oe colum Assumig that we limit the etries to real umbers, 2 3 the set of all 2 x 1 (colum) vectors is called ad the set of all 3 x 1 vectors is called Scalar multiplicatio is the process of multiplyig a vector by a real umber; the process is effected by multiplyig each etry i the vector by the scalar Vector additio ad vector subtractio are effected by addig or subtractig the correspodig etries of the two vectors; both of these operatios ca oly be preformed betwee vectors with the same umber of rows Defiitios 24 ad 25 Suppose that v1, v2,, vp are all vectors i We say that y is a liear combiatio of v1, v2,, vp if ad oly if there exist scalars, c1, c2,, cp such that y cv 1 1 cv 2 2 cpvp If such scalars exist, they are called the weights i the liear combiatio Defiitio 26 Suppose that v1, v2,, vp are all vectors i called the spa of v1, v2,, vp The set of all liear combiatios of 1 2 v, v,, vp is Theorem 21 The ceter of mass ( v ) of mass poits is: the sum of all the masses v 1 mi vi m i 1 where m i is the mass at poit v i ad m is Notes 21 ad 22 The equatio Axb has a solutio if ad oly if b ca be expressed as a liear combiatio of the colums of A; that is, Axb has a solutio if ad oly if b is i the spa of the colums of A Later i the term the spa of the colums of A will be defied as the colum space of A It follows that the equatio A x b has a solutio if ad oly if b is i the colum space of A Defiitio 28 The product of a m x matrix, A, ad x 1 vector, x, is defied by Ax xi ai where x i is the etry i the i th row of x ad a i is the i th colum of A (treatig the colums of A as vectors) You caot fid the product A x uless the umber of colums of A is equal to the umber of rows of x i 1 Vector Equatios Sectio 13/Matrix Equatios Sectio 14 23

8 MTH 261 Mr Simods class Example 31 Determie whether or ot each of the followig is a homogeous system of equatios a 2x 4x 3x 7x 5x1 6 2x b x1 x1 0 4 x x 3 x 3 0 c Example 32 Describe the solutio set to the homogeous system x x x 3 0 Example 33 Compare the solutio set from the last example with the solutio set to the ohomogeous system x x x 3 12 Example 34 Compare ad cotrast the solutio sets to the homogeous system 3x1 12x2 6x3 0 ad the ohomogeous system 3x1 12x2 6x3 15 Example 35 Show that the colum vectors of the matrix A are liearly depedet Example 36 Determie the values of a that make the colum vectors of Example 37 Prove Theorem 33 2 a 2 3 a a liearly idepedet Example 38 Determie whether or ot the colums of A spa 3 where A Homogeous Systems/Liear Idepedece Sectios 15 ad 17 31

9 MTH 261 Mr Simods class Example Cosider the set 1, 8 Explai why the set caot possibly spa vector to the set so that it does spa Example 310 Determie which of the followig sets are liearly idepedet; explai! 3 Afterwards, add a a 1 2 3,, b , 0, Defiitios ad Theorem 31 A homogeeous system of equatios is a system that ca be writte i the form Ax0 Every homogeeous system of equatios has at least oe solutio ( 0 ); 0 is called the trivial solutio to a homogeeous system of equatios Ay other solutio to a homogeous system of equatios is called a otrivial solutio Defiitios 34 ad 35 The set of vectors v 1, v 2,, v is said to be liearly idepedet if ad oly if the oly solutio to the equatio cv 1 1 cv 2 2 cv 0 is c1 c2 c 0 The set is said to be liearly depedet if there is a otrivial solutio to the equatio cv 1 1 cv 2 2 cv 0 Theorems A set of vectors cotaiig more tha vectors from must be liearly depedet A set of two vectors is liearly depedet if ad oly if oe of the vectors ca be writte as a scalar multiple of the other vector A set of vectors from spa If A is a m x matrix, the the colums of A spa row if ad oly if the set is liearly idepedet m if ad oly if A has a pivot positio i every 32 H omogeous Systems/Liear Idepedece Sectios 15 ad 17

10 MTH 261 Mr Simods class Example 41 Cosider the matrix B a What are the dimesios of the matrix? b What are etries b 32, b 23, b 41, ad b 14? Example 42 Categorize each of the followig as a vector, a colum vector, or ot a vector i ii iii 4,7, 18 iv 5 0 v Example 43 Cosider the matrices 3 1 A 5 2, B 5 1 6, ad C Fid A B, B 2C, ad C A Example 44 Which of the followig are colum vectors? T i 7 8 ii iii 4, 7, 18 T iv 5 0 v Example 45 Fid AB if 2 1 A = 4 6 ad B = Matrix Math/Iverse Matrices Sectios 21 ad

11 Example Fid AB if A = ad Example B = Fid AB if A = ad 3 2 B = Example 48 Write a simple dimesioal proof that establishes that i geeral, matrix multiplicatio is ot commutative; ie, i geeral AB BA Example 49 Use your calculator to fid I3 A ad AI 3 where A ad I Example 410 Use your calculator to fid AB where A ad B Example 411 Which of the followig is 1 C, where 5 3 C 2 1? a A b 5 3 B 2 1 c d 1 3 D M atrix Math/Iverse Matrices Sectios 21 ad 22

12 MTH 261 Mr Simods class Example 412 Determie by had if they exist, the iverses of each of the followig matrices A 4 2 4, B ad 7 12 C Example 413 Determie a geeral formula for the iverse of the matrix a A c b d Example 414 Let s use the formula derived i Example 413 to help us solve the system 4x1 3x2 20 2x1 5x2 36 Example Let A = 3 5 ad B 1, B 1, A = Fid, by had, AB, BA, AB T, T, AB 1, B A 1, A B 1 1, B 1 1 A 1, T A T, ad A 1 BA, T T A B, T T B A, A 1 See what equals what Defiitios A m matrix is a rectagular array of umbers with m rows ad colums m ad are called the dimesios of the matrix Matrices are most commoly deoted by capitol letters or sigle subscripted capitol letters umbers (etries) of a matrix are deoted by double subscripted lower case letters The Whe a matrix is explicitly writte out, it is delieated by square brackets or paretheses Abstractly, you will see thigs like A a ij where a ij represets the etry i the i th row ad j th colum Whe the umber of rows or colums goes above 9, we separate the is ad js by commas; we wo t be goig there A matrix with oly oe row is called a row vector ad a matrix with oly oe colum is called a colum vector Matrix Math/Iverse Matrices Sectios 21 ad

13 Defiitios 46 ad 47 Matrix additio: Two matrices with the exact same dimesios ca be added or subtracted thusly: a b a b ij ij ij ij If either of the correspodig dimesios of two matrices is ot the same, the matrices ca either be added or subtracted Scalar multiplicatio: A matrix ca be multiplied by a scalar (umber) thusly: k a ij ka ij Defiitio 48 The traspose of the m matrix M is the m matrix, M T, that results from swappig the rows ad colums of M Defiitio 49: Geeralized Matrix Multiplicatio If A is a m p matrix ad B is a p matrix, the the product AB is the m matrix whose ij th etry is the dot product of the i th row of A ad the j th colum of B If A is a m p matrix ad B does ot have p rows, the the product AB is ot defied Defiitios ad Theorems The matrix with etries of 1 alog the mai diagoal ad 0 i every other slot is called the idetity matrix ad is deoted as I If the products are defied, AI A ad/or I A A for ay compatible matrix A A square matrix A with dimesios is called ivertible if there exists a matrix B with the property AB I If such a matrix exists we call it the iverse of A ad symbolize it as 1 A If the A is ivertible, the 1 1 AA A A I We do ot defie iverse matrices for o square matrices 44 M atrix Math/Iverse Matrices Sectios 21 ad 22

14 MTH 261 Mr Simods class Algorithm 41 A game pla for determiig, by had, the iverse of the matrix Let s execute the pla o the ext page A is show below Let b b b We eed b31 b32 b A b21 b22 b23 1 AA I 3 which gives us: b11 b12 b b b b b31 b32 b b11 b21 b12 b22 b13 b b b b b b b b11 2b21 3b31 6b12 2b22 3b32 6b13 2b23 3b If we equate the colums of the respective matrices, we come up with three systems of three equatios with three ukows Specifically: b11 b21 1 b11 b31 0, 6b11 2b21 3b31 0 b12 b22 0 b12 b32 1, ad 6b12 2b22 3b32 0 b13 b23 0 b13 b33 0 6b13 2b23 3b33 1 A ot so close ispectio should covice you that ot oly are the coefficiet matrices for all three of the systems idetical, but they are all i fact the matrix A! Sice the row operatios performed i Gaussia elimiatio are determied solely by the coefficiet matrix, we may as well go ahead ad solve all three systems simultaeously The augmeted matrix represetatio for these three systems is: commo coefficiet matrix Costat terms, from left to right, for the first, secod, ad third colums related system of equatios Matrix Math/Iverse Matrices Sectios 21 ad

15 MTH 261 Mr Simods class Example 51 Fid a simplified formula for summig alog the secod colum a a a a first by summig alog the first row ad agai by Example 52 Use cofactors alog the secod row to fid det A where value by usig cofactors alog the first colum A Verify the determiat Example 53 Evaluate det Bwhere Example 54 Use a determiat to fid u v B where u 1, 7, 3 ad v 3, 0, Example 55: Let B = 2 1 For each of the followig matrices (each idetified as A), describe the row operatio that was affected upo I 2 to create A The fid det A ad compare its value to det I 2 Next, fid AB ad describe the differece betwee it ad B Fially, compare the values of det AB ad a A = b A c A d A 1 0 = 3 1 e det B A Example 56 Determie det A ad det B after first maipulatig the matrices ito upper triagular form A B Determiats: Sectios

16 MTH 261 Mr Simods class Example 57 Illustrate Theorem 53 usig Example 58 a b A c d ad = B x y z w Fid the area of the parallelogram outlie i Figure 1 Figure 1: A parallelogram Example 59 Use Cramer s Rule to fid the solutios to each of the followig matrix equatios a 3 1x x 2 11 b x x x 3 5 Example 510 Use the determiat ad adjoit to fid 1 A where A D etermiats: Sectios 31-33

17 MTH 261 Mr Simods class Defiitio 51: Determiats (of square matrices) det a a a For a square matrix, A, with two or more rows we defie the cofactor of etry a ij, C ij, to be 1 i j times the determiat of the matrix that results from elimiatig the i th row ad j th colum from A The usig ay row of A or ay colum of A: det A a C a C ij ij ij ij j 1 i1 Please ote that i the first formula we are summig alog a fixed i th row of A whereas i the secod formula we are summig alog a fixed j th colum of A Defiitio 52: Elemetary Matrices A elemetary matrix is a matrix that ca be created from a idetity matrix via oe elemetary row operatio Theorem 51: Elemetary Row Operatios ad Determiats Suppose that A ad B are square matrices of equal dimesio; suppose further that B ca be created from A via a sigle elemetary row operatio The: if the operatio is addig a multiple of oe row of A to a differet row of A, the det B det A if the operatio is swappig two rows of A, the det B det A if the operatio is multiplyig a row of A by the real umber k, the det B k det A Defiitios ad Theorem 52 A upper triagular matrix is a matrix where every etry below the mai diagoal is zero A lower triagular matrix is a matrix where every etry above the mai diagoal is zero The determiat of ay triagular matrix B is give by the formula det B b i 1 ii Determiats: Sectios

18 MTH 261 Mr Simods class Theorems 53 ad 54 If A ad B are like sized square matrices, the det AB det Adet B T If A is ay square matrix, the det A det A Theorem 55: A Little Geometry a a a a fids the area of ay parallelogram whose sides are parallel to a a ad a a Theorem 56: Cramer s Rule Cosider the matrix equatio Ax b where A is a osigular square matrix Defie Ai b to be the matrix derived from A by replacig the ith colum of A with b The the solutio to A x b det Ai b ca be foud usig the formula xi for each x i det A Theorem 57: A algorithm for fidig iverse matrices The matrix of cofactors of a square matrix A is the matrix that results from replacig each of its etries by their correspodig cofactors The Adjoit (Adjugate) of A is the traspose of A s matrix of cofactors The iverse of a osigular square matrix A is A Adj A det A 1 1 det A 0 This also implies, albeit less directly, that the square system Ax b has a uique solutio if ad Please ote that this implies that the matrix A is osigular if ad oly if oly if det A 0 54 D etermiats: Sectios 31-33