Mon Feb matrix inverses. Announcements: Warm-up Exercise:

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1 Math Week 6 otes We will ot ecessarily fiish the material from a give day's otes o that day We may also add or subtract some material as the week progresses, but these otes represet a i-depth outlie of what we pla to cover These otes iclude material from Mo Feb 2 35 matrix iverses Aoucemets: Warm-up Exercise:

2 Matrix iverses: A square matrix A is ivertible if there is a matrix B so that AB = BA = I, where I is the idetity matrix I this case we call B the iverse of A, ad write B = A Remark: A matrix A ca have at most oe iverse, because if we have two cadidates B, C with the AB = BA = I ad also AC = CA = I BA C = IC = C B AC = BI = B so sice the associative property BA C = B AC is true, it must be that B = C Exercise a) Verify that for A = the iverse matrix is B = Iverse matrices are very useful i solvig algebra problems For example Theorem: If A exists the the oly solutio to Ax = b is x = A b Exercise b) Use the theorem ad A i a, to write dow the solutio to the system x 2 y = 5 3 x 4 y = 6

3 Exercise 2a) Use matrix algebra to verify why the Theorem o the previous page is true Notice that the correct formula is x = A b ad ot x = b A (this secod product ca't eve be computed because the dimesios do't match up!) 2b) Assumig A is a square matrix with a iverse A, ad that the matrices i the equatio below have dimesios which make for meaigful equatio, solve for X i terms of the other matrices: XA C = B NOTE Whe A exists for a matrix A, every liear system A x = b has a uique solutio (give by the formula x = A b ) That meas that the reduced row echelo form of A must be the idetity matrix i these cases! (Because if A does't reduce to the idetity the there will be fewer tha pivots, so may systems A x = b will be icosistet ad the oes that are cosistet will have free parameters i the solutios)

4 But where did that formula for A come from? Aswer: Cosider A as a ukow matrix, A = X We wat A X = I We ca break this matrix equatio dow by the colums of X I the two by two case we get: A col X col 2 X = To be more cocrete, i this example we may write the ukow colums as col X = x x 2, col 2 X = y y 2 The we wat to solve the two systems x x 2 =, y y 2 = We ca solve for both of these mystery colums at oce, as we've doe before whe we had differet right had sides, with a double-augmeted matrix Exercise 3: Reduce the double augmeted matrix to fid the two colums of A for the previous example

5 Exercise 4: Will this always work? Ca you fid A for 5 A := ? Try to solve A X = I for the mystery matrix, ad do it colum by colum I other words, if we write x y z col X = x 2 col 2 X = x 3 y 2 y 3 col 3 X = z 2 z 3 Solve 5 x 5 y 5 z x 2 x 3 = y 2 y 3 = z 2 z 3 = with a triple-augmeted matrix reductio!

6 Exercise 5) Will this always work? Try to fid B for B := Well, if we set up the 3 this is what happes: 6 matrix where we have augmeted B with the idetity matrix, ad the reduce, What happeed, ad what does this mea? reduces to

7 Theorem: Let A be a square matrix The A has a iverse matrix if ad oly if its reduced row echelo form is the idetity I this case the algorithm illustrated o the previous page will always yield the iverse matrix explaatio: By the previous theorem, whe A exists, the solutios to liear systems A x = b are uique (x = A b So, the reduced row echelo form of A must be the idetity I i these cases I the case that A does reduce to I, we search for A as the solutio matrix X to the matrix equatio A X = I ie A col X col 2 X col X = Because A reduces to the idetity matrix, we may solve for X colum by colum as i the examples we just worked, by usig a chai of elemetary row operatios: A I I B, ad deduce that the colums of X are exactly the colums of B, ie X = B Thus we kow that A B = I To realize that B A = I as well, we would try to solve B Y = I for Y, ad hope Y = A But we ca actually verify this fact by reorderig the colums of I B to read B I ad the reversig each of the elemetary row operatios i the first computatio, i reverse order, ie create the chai B I I A So B A = I also holds! (This is oe of those rare times whe matrix multiplicatio actually is commuative) To summarize: If A exists, the solutios x to A x = b always exist ad are uique, so the reduced row echelo form of A is the idetity If the reduced row echelo form of A is the idetity, the A exists, because we have a algorithm to fid it That's exactly what the Theorem claims

8 There's a ice formula for the iverses of 2 text sectio 36 o determiats: Theorem: this case, a b c d exists if ad oly if the determiat D = ad 2 matrices, ad it turs out this formula will lead to the ext bc of a b c d is o-zero Ad i a b d b = c d ad bc c a (Notice that the diagoal etries have bee swapped, ad mius sigs have bee placed i frot of the offdiagoal terms This formula should be memorized) Exercise 6a) Check that this formula for the iverse works, for D (We could have derived it with elemetary row operatios, but it's easy to check sice we've bee haded the formula) 6b) Eve with systems of two equatios i two ukows, uless they come from very special problems the algebra is likely to be messier tha you might expect (without the formula above) Use the magic formula to solve the system 3 x 7 y = 5 5 x 4 y = 8

9 a b Remark: For a 2 2 matrix, the reduced row echelo form will be the idetity if ad oly if the c d two rows are ot multiples of each other: Ad if a, b are both o-zero the sayig that the secod row is c d ot a multiple of the first row is the same as sayig that (the ratios of the secod row etries to a b the correspodig first row etries) Cross multiplyig we see this is the same as ad bc, ie ad bc The "determiat ot equal to zero" coditio is also the correct coditio for the rows ot beig multiples, eve if oe or both of a, b are zero So this is the coditio that A reduces to the idetity ad has a iverse matrix kowig the iverse matrix exists Remark: Determiats are defied for square matrices A ad the resultig umber determies whether or ot the iverse matrices exist, (ie whether the reduced row echelo form of A is the idetity matrix) Ad, whe 2 there are aalogous (but more complicated) magic formulas for the iverse matrices whe the iverses exist, that geeralize the oe you're memorizig for = 2 This is sectio 36 material that we'll discuss carefully tomorrow, Wedesday, ad ext week Moday

10 Tues Feb 3 36 determiats Aoucemets: Warm-up Exercise:

11 Determiats are scalars defied for square matrices A They always determie whether or ot the iverse matrix A exists, (ie whether the reduced row echelo form of A is the idetity matrix): I fact, the determiat of A is o-zero if ad oly if A exists The determiat of a matrix a is defied to be the umber a ; determiats of 2 2 matrices are defied as i yesterday's otes; ad i geeral determiats for matrices are defied recursively, i terms of determiats of submatrices: Defiitio: Let A = a i j The the determiat of A, writte det A or A, is defied by det A j = a j j M j = j = a j C j Here M j is the determiat of the matrix obtaied from A by deletig the first row ad the j th colum, ad C j is simply j M j More geerally, the determiat of the matrix obtaied by deletig row i ad colum j from A is called the i j Mior M i j of A, ad C i j i j M i j is called the i j Cofactor of A Exercise Check that the messy lookig defiitio above gives the same aswer we talked about yesterday i the 2 2 case, amely a a 2 a 2 a 22 = a a 22 a 2 a 2

12 from the last page, for our coveiece: Defiitio: Let A = a i j The the determiat of A, writte det A or A, is defied by det A j = a j j M j = j = a j C j Here M j is the determiat of the matrix obtaied from A by deletig the first row ad the j th colum, ad C j is simply j M j Exercise 2 Work out the expaded formula for the determiat of a 3 3 matrix It's ot worth memorizig (as opposed to the recursive formula above), but it's good practice to write out at least oce, ad we might poit to it later a a 2 a 3 a 2 a 22 a 23 = a 3 a 32 a 33

13 Theorem: (proof is i text appedix) det A ca be computed by expadig across ay row, say row i: det A j = or by expadig dow ay colum, say colum j: a i j i j M i j = j = a i j C i j det A a i j i j M i j = a i j C i j i = i = 2 Exercise 3a) Let A := 3 Compute det A usig the defiitio (O the ext page we'll use 2 2 other rows ad colums to do the computatio)

14 From previous page, 2 A := b) Verify that the matrix of all the cofactors of A is give by C i j = The expad 5 3 det A dow various colums ad rows usig the a i j factors ad C i j cofactors Verify that you always get the same value for det A, as the Theorem o the previous page guaratees Notice that i each case you are takig the dot product of a row (or colum) of A with the correspodig row (or colum) of the cofactor matrix A := 3 C i j =

15 3c) What happes if you take dot products betwee a row of A ad a differet row of C i j? A colum of A ad a differet colum of C i j? The aswer may seem magic We'll come back to this example whe we talk about the magic formula for the iverses of 3 3 (or ) ivertible matrices A := 3 C i j =

16 Exercise 4) Compute the followig determiats by beig clever about which rows or colums to use: a) ; 4b) Exercise 5) Explai why it is always true that for a upper triagular matrix (as i 2a), or for a lower triagular matrix (as i 2b), the determiat is always just the product of the diagoal etries

17 Wed Feb 4 36 determiats midterm Friday! Aoucemets: Warm-up Exercise:

18 The effective way to compute determiats for larger-sized matrices without lots of zeroes is to ot use the defiitio, but rather to use the followig facts, which track how elemetary row operatios affect determiats: (a) Swappig ay two rows chages the sig of the determiat proof: This is clear for 2 2 matrices, sice a b c d c d = ad bc, a b = cb ad For 3 3 determiats, expad across the row ot beig swapped, ad use the 2 2 swap property to deduce the result Prove the geeral result by iductio: oce it's true for matrices you ca prove it for ay matrix, by expadig across a row that was't swapped, ad applyig the result (b) Thus, if two rows i a matrix are the same, the determiat of the matrix must be zero: o the oe had, swappig those two rows leaves the matrix ad its determiat uchaged; o the other had, by (a) the determiat chages its sig The oly way this is possible is if the determiat is zero

19 (2a) If you factor a costat out of a row, the you factor the same costat out of the determiat Precisely, usig for i ith row of A, ad writig = c * i i : i = c : i * = c : i * proof: expad across the i th row, otig that the correspodig cofactors do't chage, sice they're computed by deletig the i th row to get the correspodig miors: det A = j = a i j C i j = j = * c a Ci i j j = c j = * a Ci i j j = c det A * (2b) Combiig (2a) with (b), we see that if oe row i A is a scalar multiple of aother, the det A =

20 (3) If you replace row i of A, i by its sum with a multiple of aother row, say determiat is uchaged! Expad across the i th row: k the the i c 2 k k = j = a c a C = a C i j k j i j j = i j i j c a C = det A j = k j i j c 2 k k = det A Remark: The aalogous properties hold for correspodig "elemetary colum operatios" I fact, the proofs are almost idetical, except you use colum expasios

21 2 Exercise ) Recompute 3 from yesterday (usig row ad colum expasios we always got 2 2 a aswer of 5 the) This time use elemetary row operatios (ad/or elemetary colum operatios) 2 Exercise 2) Compute 2 2 2

22 Theorem: Let A The A exists if ad oly if det A proof: We already kow that A exists if ad oly if the reduced row echelo form of A is the idetity matrix Now, cosider reducig A to its reduced row echelo form, ad keep track of how the determiats of the correspodig matrices chage: As we do elemetary row operatios, if we swap rows, the sig of the determiat switches if we factor o-zero factors out of rows, we factor the same factors out of the determiats if we replace a row by its sum with a multiple of aother row, the determiat is uchaged Thus, A = c A = c c 2 A 2 = = c c 2 c N rref A where the ozero c k 's arise from the three types of elemetary row operatios If rref A = I its determiat is, ad A = c c 2 c N If rref A I the its bottom row is all zeroes ad its determiat is zero, so A = c c 2 c N = Thus A if ad oly if rref A = I if ad oly if A exists! Remark: Usig the same ideas as above, you ca show that det A B = det A det B This is a importat idetity that gets used, for example, i multivariable chage of variables formulas for itegratio, usig the Jacobia matrix (It is ot true that det A B = det A det B ) Here's how to show det A B = det A det B : The key poit is that if you do a elemetary row operatio to AB, that's the same as doig the elemetary row operatio to A, ad the multiplyig by B With that i mid, if you do exactly the same elemetary row operatios as you did for A i the theorem above, you get A B = c A B = c c 2 A 2 B = = c c 2 c N rref A B If rref A = I, the from the theorem above, A = c c 2 c N, ad we deduce A B = A B If rref A I, the its bottom row is zeroes, ad so is the bottom row of rref A B Thus A B = ad also A B =

23 There is a "magic" formula for the iverse of square matrices A (called the "adjoit formula") that uses the determiat of A alog with the cofactor matrix of A We'll talk about the magic formula o Moday ext week, after the midterm

24 Exam otes: The exam is this Friday February 6, from :4-:4 am Note that it will start 5 miutes before the official start time for this class, ad ed 5 miutes afterwards, so you should have oe hour to work o the exam Get to class early, ad brig your Uiversity ID card, which we might ask you to show if we do't recogize you from sectios or lecture This exam will cover textbook material from -5, 2-24, 3-35 The exam is closed book ad closed ote You may use a scietific (but ot a graphig) calculator, although symbolic aswers are accepted for all problems, so o calculator is really eeded (Usig a graphig calculator which ca do matrix computatios for example, is grouds for receivig grade of o your exam So please ask before the exam if you're usure about your calculator Ad of course, your cell phoes must be put away) I recommed tryig to study by orgaizig the coceptual ad computatioal framework of the course so far Oly the, test yourself by makig sure you ca explai the cocepts ad do typical problems which illustrate them The class otes ad text should have explaatios for the cocepts, alog with worked examples Old homework assigmets ad quizzes are also a good source of problems It could be helpful to look at quizzes/exams from my previous Math 225 classes, which go back several years from the lik Your lab meetigs tomorrow will be exam review sessios

25 Chapter 3: 3a) Ca you recogize a algebraic liear system of equatios? 3b) Ca you iterpret the solutio set geometrically whe there are 2 or three ukows? 3c) Ca you use Gaussia elimiatio to compute reduced row echelo form for matrices? Ca you apply this algorithm to augmeted matrices to solve liear systems? 3d) What does the shape of the reduced row echelo form of a matrix A tell you about the possible solutio sets to Ax = b (perhaps depedig, ad perhaps ot depedig o b )? Focus especially o whether each row of the reduced matrix has a pivot or ot; ad o whether each colum of the reduced matrix has a pivot or ot 3e) What properties do (ad do ot) hold for the matrix algebra of additio, scalar multiplicatio, ad matrix multiplicatio?

26 3f) What is the matrix iverse, A for a square matrix A? Does every square matrix have a iverse? How ca you tell whether or ot a matrix has a iverse, usig reduced row echelo form? What's the row operatios way of fidig A, whe it exists? Ca you use matrix algebra to solve matrix equatios for ukow vectors x or matrices X, possibly usig matrix iverses ad other algebra maipulatios?

27 Chapters -2: a) What is a differetial equatio? What is its order? What is a iitial value problem, for a first or secod order DE? b) How do you check whether a fuctio solves a differetial equatio? A iitial value problem? c) What is the coectio betwee a first order differetial equatio ad a slope field for that differetial equatio? The coectio betwee a IVP ad the slope field? d) Do you expect solutios to IVP's to exist, at least for values of the iput variable close to its iitial value? Why? Do you expect uiqueess? What ca cause solutios to ot exist beyod a certai iput variable value? e) What is Euler's umerical method for approximatig solutios to first order IVP's, ad how does it relate to slope fields? f) What's a autoomous differetial equatio? What's a equilibrium solutio to a autoomous differetial equatio? What is a phase diagram for a autoomous first order DE, ad how do you costruct oe? How does a phase diagram help you uderstad stability questios for equilibria? What does the phase diagram for a autoomous first order DE have to do with the slope field?

28 g) Ca you recogize the first order differetial equatios for which we've studied solutio algorithms, eve if the DE is ot automatically give to you pre-set up for that algorithm? Do you kow the algorithms for solvig these particular first order DE's? 2) Ca you covert a descriptio of a dyamical system i terms of rates of chage, or a geometric cofiguratio i terms of slopes, ito a differetial equatio? What are the models we've studied carefully i Chapters -2? What sorts of DE's ad IVP's arise? Ca you solve these basic applicatio DE's, oce you've set up the model as a differetial equatio ad/or IVP?