Teaching and Learning Guide 10: Matrices

Transcription

1 Teching nd Lerning Guide : Mtrices

2 Teching nd Lerning Guide : Mtrices Tle of Contents Section : Introduction to the guide. Section : Definitions nd Opertions.. The concept of definitions nd opertions. Presenting the concept of definitions nd opertions.5. Delivering the concept of definitions nd mtri opertions nd to smll or lrger groups..6. Discussion Questions.. 5. ctivities. 6. Top Tips. 7. Conclusion Section : Trnsposing nd Inverting Mtri nd Mtri Determinnts. The concept of trnsposition, inversion nd mtri determinnts.. Presenting the concept of trnsposition, inversion nd mtri determinnts6. Delivering the concept of trnsposition, inversion nd mtri determinnts to smll or lrger groups.8. Discussion Questions ctivities.9 6. Top Tips.9 7. Conclusion Section : Crmer s Rule... The concept of Crmer s rule... Presenting the concept of Crmer s rule. Delivering the concept of Crmer s rule to smll or lrger groups.. Discussion Questions ctivities.5 6. Top Tips.8 7. Conclusion 8 Section 5: Input-Output nlysis. 8. The concept of input-output nlysis8. Presenting the concept of input-output nlysis.9. Delivering the concept of input-output nlysis to smll or lrger groups.. Discussion Questions.. 5. ctivities. 6. Top Tips.5 7. Conclusion 5 Pge of 5

3 Teching nd Lerning Guide : Mtrices Section : Introduction to the guide This guide is designed to set out some of the sic mthemticl concepts needed to tech economics nd finncil economics t undergrdute level. The concepts covered y this guide re (i) the dimensions of mtri nd surrounding voculry; (ii) ddition, sutrction, multipliction nd division of mtrices; (iii) mtri trnsposition; (iv) mtri inversion; (v) finding the determinnt of mtri; (vi) Crmer's rule; (vii) Input-Output nlysis. It is very useful to use Ecel to ssist teching the topic of mtrices. Ecel hs lrge numer of in uilt functions to help find the trnspose nd inverse of mtrices. It lso hs n inuilt function to multiply mtrices. One key issue in mtri multipliction is conformility. Ecel focuses on conformility directly s efore you undertke ny mtri opertions in Ecel you need to determine the dimension of the resultnt mtri nd highlight selection of cells mtching this dimension. If you highlight n incorrect dimension Ecel is unle to undertke the clcultion. The use of Ecel is n essentil tool for nyone working in finnce. Throughout this guide Ecel screenshots nd links to files re provided. It would e useful therefore if the session utilising this mteril were presented in clssroom where students cn gin hnds on eperience. Mtrices re commonly used in finnce. s consequence numer of the emples hve finnce is. These include (i) using mtrices to clculte covrince mtri; (ii) using mtrices to clculte the risk of shre portfolio. n emple of how mtrices re used in journl rticle is included s teching nd lerning ctivity. This is n ecellent wy of demonstrting to students tht lerning mthemticl techniques is not simply cse of lerning for the ske of lerning. It is not lwys possile to find pproprite emples in journl rticles ut the one included in this guide is set t suitle level. The lecturer is lso directed to n lterntive rticle for n eercise tht could e used s tutoril or emintion question. With the use of Ecel for mtri multipliction nd inversion it is less pprent on the reltive dvntge of using Crmmers rule over stndrd techniques to find solutions to prolems. n lgeric sed emple is included to show tht Crmmers rule is still useful. This topic is Pge of 5

4 Teching nd Lerning Guide : Mtrices most definitely doing topic. Consequently lrge numer of emples re included to help the lecturer. Section : Definitions nd Opertions. The concept of definitions nd opertions Mtrices re difficult topic for mny students nd set of cler definitions re very importnt. These will need to e revisited to ensure students hve secure understnding of the key terms. Some definitions tht might e useful include: ) Defining mtri mtri is rectngulr rry of numers, prmeters or vriles rrnged in some meningful order. The elements (or prmeters or vriles) re referred to s the elements of mtri. The elements in horizontl line constitute row of the mtri nd it follows tht the elements in verticl line constitute column of the mtri. The entries in mtri re usully enclosed in two curved lines or squre rckets. Thus the generl mtri with m rows nd n columns cn therefore e written s: m m. n n mn The element in the i th row nd the j th column is ij. If we cll the mtri ove,, we cn sometimes void writing the mtri out in full, nd insted write, very succinctly,. ) Defining the dimensions mtri, like the one ove, with m rows nd n columns is clled n m y n or n m n mtri. This determines the dimensions of the mtri. Lecturers could remind students tht in our emple tht m is the numer of rows nd n is the numer of columns nd lso to e cler tht the row numer lwys precedes the column numer. Pge of 5

5 Teching nd Lerning Guide : Mtrices. Presenting the concept of definitions nd opertions It is esy to present mtrices s purely strct nd theoreticl concept. The dnger of such n pproch is tht the mthemtics cn e perceived y students to ecome nd end in itself; tht the point of lerning out mtrices is simply to prctise methodology nd pss n em. The most effective presenttions of mtrices will continully contetulise mtrices nd give rel world emples to support the conceptul frmework. In some wys, with hrd topic like mtrices the lecturer might even find tht there is rel impertive to put mtrices into n pplied setting lmost s wy to convince students tht it is vlid, tht mtrices re legitimte tool which cn inform economic prolems. For emple, the following emple is intended to show tht mtrices re compct wy of rticulting mthemticl prolems. Consider the risk (mesured y vrince) of two sset stock portfolio cn e written s: σ where : σ σ p σ σ is the proportion of welth invested in sset, is the proportion of welth invested in sset, is the vrince of sset, σ is the vrince of sset, σ is the covrince etween ssets nd. This formul cn e etended to, or n ssets. However s we etend the numer of ssets the numer of terms gets lrger. In prticulr if we hve n ssets there n(n-)/ covrince terms to write down, i.e. n there re 99/ 95 covrince terms. Therefore epressing this eqution in its liner form cn ecome prolemtic s the numer of ssets gets lrge. Pge 5 of 5

6 Teching nd Lerning Guide : Mtrices However this eqution cn e generlised to m ssets nd written more compctly using σ p ' where : mtrices. is vector of portfolio weights (welth invested), is mtri of covrinces etween ssets Hopefully, students will gree tht the second representtion is esier to rememer!. Delivering the concept of definitions nd mtri opertions nd to smll or lrger groups In terms of delivering the nuts nd olts of sic mtri opertion, there is little sustitute for ctully going through the methods with students either in lecture or tutoril setting. Students will need to see cler emples of ddition, sutrction, etc., nd rief overview is provided elow to help collegues deliver this introductory mteril. This is not intended to replce good tetook pproch ut could inform set of PowerPoint slides or lecture hndout. Mtri opertions ddition: Remind students tht with regulr lger we could write c, wheres with mtri lger we would write C B. They will need to know tht in order to do this, mtrices nd B must e of the sme dimension. If this is the cse then ech element of one mtri is dded to the corresponding element of the other mtri. Thus if nd B re oth mtrices element will e dded to, dded to, dded to nd dded to. ddition: Worked Emple 7 9, B B Pge 6 of 5

7 Teching nd Lerning Guide : Mtrices Sutrction: gin, students could look t regulr lger where we could write c -, wheres with mtri lger we would write C - B. However in order to do this, mtrices nd B must e of the sme dimension. If this is the cse then ech element of one mtri is sutrcted from the corresponding element of the other mtri. Thus if nd B re oth mtrices element will e sutrcted from to, sutrcted from, sutrcted from nd sutrcted from. Sutrction: Worked Emple 7 9, B B Sclr Multipliction Students will need to e cler tht in liner lger rel numer such s 9, - or.6 is clled sclr. Moreover, they will need to see tht the multipliction of mtri y sclr involves multipliction of ech element of the mtri y the sclr nd tht this type of multipliction is esy to rememer s it scles the mtri up or down ccording to the vlue of the mtri. Sclr Multipliction: Worked Emple 7, k k 6 The Ecel file contining the solutions to these three prolems cn e found online t Pge 7 of 5

8 Teching nd Lerning Guide : Mtrices Lecturers might wnt to flg up to students tht in order to undertke mtri ddition, sutrction nd sclr multipliction in Ecel you must first of ll define the nme of the mtri or vector. For emple, in this spredsheet we must first of ll highlight cells B:C then select Insert, Nme, Define nd cll this rnge of cells. Repet for B. Then when it comes to dding to B, the result of which will lso e mtri, you must first of ll highlight lock of cells nd then type B. But rther thn pressing Enter you must press CTRLSHIFTENTER together to otin the nswer. Multipliction It is often useful strting point to stte tht it is often esiest with mtri or vector multipliction to sk yourself wht the nswer will e first of ll. Students will need to know tht mtri multipliction in Ecel requires this s you hve first of ll highlight the cells where the nswer will e displyed. Students could e referred ck to the originl emple where the dimensions of mtri were stted s m n where m is the numer of rows nd n is the numer of columns. The net Pge 8 of 5

9 Teching nd Lerning Guide : Mtrices step could e to eplin tht to multiply (m n) mtri y nother (n m) then the resultnt mtri would e m m nd the multipliction opertion hs only een possile since the numer of columns in the first mtri (the led mtri) is equl to the numer of columns in the second mtri (the lg mtri). Hint: suggestion s wy to rememer this mde y Dowling (99) is to plce the two sets of dimensions in the order in which they re to e multiplied. Then circle (mentlly or physiclly) the lst numer of the led mtri dimensions nd the first numer of the lg mtri dimensions. If the two numers re equl then they re conformle (i.e. mtri multipliction is possile in the given order). Furthermore, the outer numers will provide the dimensions of the product mtri. Multipliction: Worked emple Below we consider the multipliction of mtri y mtri B, i.e. B. Dimensions of mtri Dimensions of mtri B Is B possile? ( ) ( ) Yes, numer of columns in () equls the numer of rows in B () ( ) ( ) Yes, numer of columns in () equls the numer of rows in B () ( ) ( ) Yes, numer of columns in () equls the numer of rows in B () (5 ) ( 5) Yes, numer of columns in () equls the numer of rows in B () ( ) ( ) Yes, numer of columns in () equls the numer of rows in B () No, numer of columns in () is ( ) ( ) not equl to the numer of rows in B () Dimensions of resultnt mtri ( ) - rows nd columns ( ) - rows nd columns ( ) - sclr ( 5 5) - 5 rows nd 5 columns ( ) - rows nd column N/ Pge 9 of 5

10 Teching nd Lerning Guide : Mtrices ( ) ( ) (5 ) (5 ) No, numer of columns in () is not equl to the numer of rows in B () No, numer of columns in () is not equl to the numer of rows in B (5) N/ N/ Links with the online question nk Generl questions on mtrices cn e found t: nd questions focused specificlly on multipliction re lso locted t: Discussion Questions Students could e sked to reserch some usiness or economics dt which they present s simple mtri e.g. shre dt, GDP figures over time etc 5. ctivities CTIVITY ONE Lerning Ojectives LO: Students lern how to complete mtri multipliction LO: Students lern how to use Ecel to complete mtri multipliction LO: Students lern how to use the mtri opertors Tsk One If: 6 nd B Clculte () B nd () B. Pge of 5

11 Teching nd Lerning Guide : Mtrices Tsk Two n ecellent wy of reinforcing students understnding of mtri multipliction is to use Ecel. The pproprite function is MMULT. There re two key differences when using this function compred to the mjority of Ecel functions: Rther thn pressing Enter or clicking OK if using the function wizrd you must press CTRL Shift Enter together to tell Ecel to undertke the clcultions. Since the outcome of this function is usully going to e something other thn sclr you must inform Ecel where to disply the output. Thus if the outcome is ( ) mtri you must first of ll highlight rows nd column efore strting to use the function. Students might wish to see video showing these opertions eing undertken using Ecel Tsk Three If: nd B Clculte B. NSWERS CTIVITY ONE Tsk One Rememering wht we hve lerned previously nd following the helpful hint: ( ) ( ) ( ) Therefore whichever wy we perform the multipliction the result will e mtri. Pge of 5

12 Teching nd Lerning Guide : Mtrices Pge of B B Tsk Two - Tsk Three B 6. Top Tips Tip : Specil cses Lecturers will wnt to ensure tht the specil cses re fully understood. Tht is: squre mtri in squre mtri the numer of rows equls the numer of columns. squre mtri differs from other mtrices in tht it hs digonls. The digonl leding from top left to ottom right is clled the min digonl. squre mtri which hs ll its elements zero ecept for those long its min digonl, nd these re ll ones, is clled unit mtri or identity mtri. mtri composed of severl rows nd single column hs dimensions m nd is referred to s column vector. mtri composed of single row nd severl columns hs dimensions n nd is referred to s row vector. This could e presented in simple templte or perhps s spider digrm. Tip : Multipliction When pproching this topic is fr esier to let students ecome cquinted with multiplying vectors efore they move onto multiplying mtrices. Before ttempting mtri multipliction students should consider will the dimensions of the resultnt mtri e?

13 Teching nd Lerning Guide : Mtrices 7. Conclusion This preprtory work will e crucil to the susequent success students chieve with higher level nd more comple mtri work. ny scheme of work should ensure tht plenty of time is given for students to prctise these sic opertions nd for one-to-one support to e given to remedy ny initil prolems. Section : Trnsposing nd Inverting Mtri nd Mtri Determinnts. The concept of trnsposition, inversion nd mtri determinnts s noted erlier, this is topic which students will perceive s difficult. This is compounded y the rther opque terminology used nd one key responsiility incument upon lecturers is to help students mke sense of these lels nd concepts. ) Trnsposition For emple to show tht the term trnsposition is ctully something quite simple s: If the rows nd columns of mtri re interchnged the new mtri is known s the trnspose of the originl mtri. E.g. If then the trnspose, ' This could even e introduced using imges rther thn numers such s: If Z Pge of 5

14 Teching nd Lerning Guide : Mtrices Then Z nd then more formlly tht If then ' ) Determinnts Similrly, lecturers could use simple ut forml overview y stting tht the determinnt is squre group of numers set out in mtri formt ut enclosed y stright rther thn curved Pge of 5

15 Teching nd Lerning Guide : Mtrices lines. It is importnt tht students hve mstered find the determinnt of mtri efore progressing to nd ove. The determinnts of orders nd re defined s: For emple, finding the vlue of the following determinnts of order : 5 5 ( ) ( )( ) c) Inversion Similrly, suppose we hve mtri : c d Then the inverse of is: d c Hence to otin the inverse of squre mtri of order we: Step : interchnge the elements on the min digonl, Step : chnge the sign of the other two elements, Pge 5 of 5

16 Teching nd Lerning Guide : Mtrices Step : divide y the determinnt corresponding to the originl mtri. This is simple teching sequence of core concept ut one through which students will need to e guided crefully.. Presenting the concept of trnsposition, inversion nd mtri determinnts Trnspositions re firly strightforwrd to tech nd lecturers might wnt to focus on inversions nd mtri determinnts. n outline of mteril tht could e presented is set out elow Inversions Inversions re something which students need to prctise. Students cn e reminded tht in trditionl rithmetic division is the reciprocl or inversion of multipliction. If we consider sy divided y, to tke simple emple, we sy tht ¼ nd ¼. In mtri lger we define inversion using the sme pproch. We sy tht the inverse of squre mtri is nother squre mtri B where: B I nd B I where I is the identity mtri of the sme order. Such mtri B, if it eists, is unique nd is clled the inverse of (written s - ). Consequently we hve: - I nd - I Students could consider these two squre mtrices: 7 nd 7 - Pge 6 of 5

17 Teching nd Lerning Guide : Mtrices Pge 7 of 5 Multiplying these two together we get: 7 ) ( ) ( 7 7 ) ( ) ( Or, 7 ) ( ) ( 7 ) ( 7 ) ( Clerly these two mtrices re inverses of one nother nd students need to understnd this technique Determinnts Students will need to e wre of the process for clculting determinnts including the notion of co-fctors. gin, some useful mteril is provided elow which cn e incorported into notes or slides. discussion of the determinnts of mtri will need to first introduce the concept of cofctors. Students could e shown tht Corresponding to ech element ij, of mtri,, there is cofctor, ij. nd tht it follows tht mtri hs nine elements, so there re nine cofctors, to e computed. The cofctor, ij, is defined to e the determinnt of the mtri otined y deleting row i nd column j of, prefied y either sign or s - sign. The prefies re ssigned ccording to the following pttern:

18 Teching nd Lerning Guide : Mtrices For emple, suppose we wish to clculte, which is the cofctor ssocited with in the mtri: The element lies in the third row nd second column. Consequently we delete the third row nd the second column to produce the mtri: The cofctor, is the determinnt of this mtri prefied y - sign due to the ssigning pttern ove, i.e. ( ) Now we hve introduced the cofctor concept we cn move onto showing the determinnt of mtri. The determinnt of the generl mtri is s follows: Students should note tht every one of these si products of elements contins ectly one element from ech row nd ectly one element from ech column.. Delivering the concept of trnsposition, inversion nd mtri determinnts to smll or lrger groups Lecturers could deliver this mteril through series of forml lectures ut supplement with student-led tutorils. For emple, students could e invited to sign-up to deliver mini-lectures Pge 8 of 5

19 Teching nd Lerning Guide : Mtrices covering one or more these topics nd other students re required to sign-up to ttend t lest two s udience memers. This cn e simple ut effective technique to help students grsp the essentils of mtri lger; if student cn deliver concise nd cogent overview then they re very likely to hve understood the mteril. This teching strtegy could e differentited y piring higher ility with lower ility students to led ech workshop. Links with the online question nk Questions on determinnts cn e found on the METL wesite t: whilst tsks centred on mtri inversion re ville from: Discussion Questions Students could e encourged to ring ny prolem sets they hve een working on into tutoril nd to discuss ny prolems they hve encountered. This could identify to the lecturer or tutor ny common misunderstndings or res of difficulty nd inform the tutor s tutoril scheme of work. This co-construction of future teching nd lerning ojectives is n importnt prt of successful ssessment for Lerning strtegy. 5. ctivities CTIVITY ONE Lerning Ojectives LO: Students hve opportunity to refresh clcultion of verges LO: Students to lern how to clculte verges using mtrices LO: Students lern how mtrices cn e used in pplied economics Consider the dt elow which re the weekly stock price returns of three UK gmling stocks: SPORTINGBET 888 HOLDINGS PRTYGMING 7/7/ /7/ /7/ Pge 9 of 5

20 Teching nd Lerning Guide : Mtrices 7/8/ /8/ /8/ /8/ /9/ /9/ /9/ /9/ // The verge weekly return is: Sporting Bet -8.7% 888 Holdings -.8% Prtygming -.8% Tsk One Using mtrices how would I find the verge return of Sporting Bet? Tsk Two Confirm tht the verge of oth 888 Holdings nd Prtygming is -.8% CTIVITY TWO Lerning Ojectives LO: Students lern the mening nd significnce of covrince LO: Students lern how to clculte vrince using mtri lger LO: Students lern how to pply mtri lger to solve n pplied economics prolem LO: Students lern how to mke n economic inference using covrince informtion Bckground informtion The vrince of portfolio cn e clculted s: Pge of 5

21 Teching nd Lerning Guide : Mtrices σ p ' where : is vector of portfolio weights (welth invested), is mtri of covrinces etween ssets For four sset cse this would look like the following when epnded out: σ p where : i ( ) is the weight invested in sset i, σ is the covrince etween sset i nd j, ij σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ Note here we first of ll multiply () row vector y () mtri to otin () row vector which is multiplied y () column vector to give () nswer. Tsk One Let us ssume the following covrinces etween four UK stocks: Covrince MMO nd Mrks Spencer Brclys Scottish Power MMO Mrks nd Spencer Brclys Scottish Power Clculte the vrince of n eqully weighted ( i.5) portfolio using mtri lger Tsk Two Wht cn you infer out risk? Pge of 5

22 Teching nd Lerning Guide : Mtrices CTIVITY THREE Lerning Ojectives LO: Students lern how to find the inverse of mtri LO: Students lern how to clculte the determinnt Tsk One Find the inverse of the following mtri 5 Tsk Two Find the determinnt of the following mtri: 5 9 Tsk Three Find the inverse of the following mtri: CTIVITY FOUR Lerning Ojectives LO: Students lern how to independently clculte determinnts LO: Students lern how to mke inferences out whether mtri is singulr or not LO: Tsk One Find the determinnt of the following mtri: Pge of 5

23 Teching nd Lerning Guide : Mtrices Tsk Two Find the determinnt of the following mtri: 9 NSWERS CTIVITY ONE Tsk One n verge is found y dding up series of numers nd dividing y the numer tht they re. So for the series ove there re twelve numers so we dd them together nd divide y. To replicte the figure of -8.7% ove using mtrices we would multiply:.8 ( ) Tsk Two Students should confirm through mtri lger tht the ssertion is indeed correct. Pge of 5

24 Teching nd Lerning Guide : Mtrices CTIVITY TWO Tsk One σ p ( ) Note tht the digonl elements here re the individul vrinces of the four stocks. We cn see tht MMO is the most voltile stock. Interestingly, we lso see tht MMO nd Scottish Power hve negtive covrince. Using the MMULT commnd in Ecel we cn find the nswer very esily: ' ' ' 5.56 Pge of 5

25 Teching nd Lerning Guide : Mtrices Tsk Two Notice tht the nswer of 5.56 is less thn ny of the individul vrinces nd hence we cn reduce risk y putting 5% of our welth in ech sset rther thn putting ll our eggs in one sket. This spredsheet is ville t Note when crrying out these clcultions using Ecel it is possile to nest functions s follows: Note: Rther thn referring to n rry of cells s sy B:E (s in the screen shot ove) it is possile to nme the rnge of cells (Using Insert, Nme, Define ). You could therefore replce B:E with sigm in the emple ove. Pge 5 of 5

26 Teching nd Lerning Guide : Mtrices CTIVITY THREE Tsk One Tsk Two Epnding y the rd row (due to the presence of zero): ( 9) ( 5 ) 6 8 Tsk Three Tsk Three First of ll finding the determinnt y epnding y column : det() (6-9) ( 9) (9-) Then finding the nine cofctors: (6 9) 7 ( 9) (9 ) ( ) ( ) ( ) ( ) ( ) ( ) Putting these in-order forms the djugte mtri: Pge 6 of 5

27 Teching nd Lerning Guide : Mtrices 7 Then tking the trnspose of the djugte mtri to form the djoint mtri: 7 This ws very stright forwrd tsk due to the symmetry of the djugte mtri. The inverse is therefore: 7 7 Why hve we done ll this? Hopefully, students will e questioning why there hs een so much emphsis on finding the determinnts nd inverses of mtrices. In order to reinforce tht it is time well spent we should use numer of emples. CTIVITY FOUR Tsk One Epnding y the rd row (due to the presence of zero): ( 5 5) ( 5 5) 5 5 The vlue of the determinnt is zero. This implies tht the corresponding mtri is singulr nd hence hs no inverse. Pge 7 of 5

28 Teching nd Lerning Guide : Mtrices Pge 8 of 5 Tsk Two The first thing to sk if the pttern of s nd - s. This is just n etension of the pttern introduced erlier: The net thing to consider is wht row or column is it optiml to epnd y. If we look closely we will see tht row nd column oth hve two zero s which will reduce the numer of clcultion. Therefore we should choose either of those. Epnding y column : 9 We now hve to find the determinnts of the cofctor mtrices. Consider the term efore the minus sign nd epnding y row : ( ) ( ) { } ( ) { } ( ) Now considering the term fter the minus sign nd epnding y row : 9 9 ( ) ( ) { } 9 9

29 Teching nd Lerning Guide : Mtrices { ( 5) ( ) } { } The determinnt of this mtri is therefore Top Tips Tip : Demonstrte to students tht It is lso possile to find the inverse of mtrices using Ecel. gin you must highlight the cells where you would like the result to disply. The pproprite function is MINVERSE. Pge 9 of 5

30 Teching nd Lerning Guide : Mtrices Tip : Students might find it helpful to think of cofctors using simple shded digrm nd to then see the pttern ecomes evident. In the tle elow we strt in the top left cell (representing element ) nd then highlight cell in different row nd different column ( ). By defult the only remining cell to choose tht is from different column nd different row is the ottom right (representing element ). We then strt gin with the sme top left cell highlighted ut select different second row cell ( ) which in turn forces us to select s the third row cell. We continue this process until we hve ehusted ll the potentil comintions tht hve cell from different column nd row highlighted. The lgeric epression ove cn e fctored out to ecome: ( ) ( ) ( ) Hopefully we recognised tht the terms inside the rckets re the cofctors of the elements outside the rckets. We should lso recognise the pttern of s nd - s from the tle ove. Pge of 5

31 Teching nd Lerning Guide : Mtrices Tip : With mny topics in mthemtics there eists more thn one method to find the solution to prolem. One method populr with students for finding the determinnt of mtri is s follows (using the emple ove): 5 9 shortcut involves dding copy of the first nd second rows s follows: nd then multiplying s follows: sum 7 nd: sum9 5 9 The difference etween these is then 7-9 8, which s we found erlier is the determinnt of the mtri. You could sk the students why this works. The nswer of course is tht this is simply n lterntive fctoristion of the epression given ove. Pge of 5

32 Teching nd Lerning Guide : Mtrices 7. Conclusion Crete opportunities in scheme if work for students to work through prolem sets in pirs. This cn e helpful tool to rise ny prolems or queries tht students might hve. This topic drws upon nd pulls together mny different ut complementry threds nd students re likely to enefit from this collortion. Section : Crmer s Rule. The concept of Crmer s rule simple sttement t the eginning of lecture will prove helpful. Perhps: The method offered y Crmer s rule llows us to finds the vlue of one vrile t time nd this method requires less effort if only selection of the vriles is required. Crmmers rule for solving ny n n system,, sttes tht the i th vrile, i, cn e found from: i i Where i is the n n mtri formed y replcing the i th column of y the right-hnd side vector,.. Presenting the concept of Crmer s rule Students will wnt to see simple presenttion of how Crmer s rule works in prctice. One strtegy for doing this is set out elow: Consider the simple system considered previously: 7 P Q 9 nd suppose tht we need to find the vlue of the second vrile, Q. Pge of 5

33 Teching nd Lerning Guide : Mtrices ccording to Crmmers rule: Q Where 7 nd hence nd nd hnce 79 ( ) Therefore Q -5/-6. Delivering the concept of Crmer s rule to smll or lrger groups worked emple is offered elow to help collegues delivering the concept. Smller groups will proly hve more scope to discuss ech step outlined: Worked emple of Crmer s rule Consider the following three interconnected mrkets: Mrket : Q S 6P 8 nd Q D -5P P P Mrket : Q S P nd Q D P -P P 5 Mrket : Q S P 5 nd Q D P P - P 9 These mrkets re simultneously in equilirium when quntity supplied equls quntity demnded in ll three mrkets: Mrket : 6P 8-5P P P Mrket : P P -P P 5 Mrket : P 5 P P - P 9 Rerrnging these epressions so tht they cn e written in mtri form: Pge of 5

34 Teching nd Lerning Guide : Mtrices Mrket : P - P - P Mrket : -P 6P - P 6 Mrket : - P -P 7P Hence, in mtri form: 6 P P 6 7 P Or. Using the mtri inversion pproch we would e required to invert the mtri, which involves finding its determinnt nd the nine cofctors. Using Crmmers rule we still hve to find the determinnt of mtri ut we then only hve to find the determinnt of the three i mtrices P, P 7, P Or using the inversion pproch: Pge of 5

35 Teching nd Lerning Guide : Mtrices Students should note however the reltive merits of Crmmers rule over the inversion pproch only pplies to mnul clcultions. The screen shot elow shows how quick it ws to find the solutions to these prolems simply y using the inversion pproch. The file is ville online t: Links with the online question nk See Discussion Questions Students could discuss why Crmer s rule offers n dvntge in terms of clcultion. 5. ctivities Lerning Ojectives LO: Students lern the prcticl ppliction of Crmer s rule to ntionl income determintion Pge 5 of 5

36 Teching nd Lerning Guide : Mtrices LO: Students pply their knowledge to simple prolem of evluting ntionl income Tsk One Given the following typicl Ntionl Income model found in most introductory mcroeconomics tet ooks: Y C I G C Y d T ty Where Y Income, Y d disposle income, C Consumption, TTtion, mrginl propensity to consume, t rte of ttion, I utonomous Investment nd G utonomous Government spending. Hence: Re-writing the three equtions in terms of Y, C nd T we otin: Y C I G C (Y-T) Y T nd so Y C T -ty T These three equtions cn e represented in mtri form s: Y I G C t T Using Crmmers rule we cn solve for the equilirium levels of Y nd/or C nd/or T s follows: Pge 6 of 5

37 Teching nd Lerning Guide : Mtrices Pge 7 of 5 ( ) t G I t t G I t G I Y ) ( ) ( t t G I t t t t G I t t G I C ) ( ) ( ) ( ) ( t G t I t t t G I t t t G I T ) ( ) ( ) ( ) ( We therefore hve lgeric solutions tht llow us to consider wht would hppen to the equilirium levels of Y, C nd T if, for emple, the Government chnged G or t. Tsk One () Review the ove eposition of Crmer s rule. () ssume tht I, G,.7 nd t.. Wht is the vlue of Y?

38 Teching nd Lerning Guide : Mtrices NSWER Tsk One () I G 5 5 Y 6.6 t ( I G) ( t) ( ).7 (.8) 5.56 C 66.6 t.. ( I G) t ( ). T 7.7 t.. Y C I G Top Tips sk students to use Ecel wherever they cn to rticulte nd solve mtri lger prolems. Cn they show their nswer to 5. (see ove) using Ecel? 7. Conclusion Students could e sked to crete their own plenry s prt of conclusion on Crmer s rule. For emple, how does Crmer s rule help us s economists? Wht sort of economic issues might it e prticulrly pplicle to? This evlution of Crmer s rule could e good wy to conclude this topic. Section 5: Input-Output nlysis. The concept of input-output nlysis This mteril could e prefced y eplining tht in clssicl economic nlysis it is ssumed tht country s economic ctivity cn e divided into numer of industries which produce goods nd services. In ddition, it is further ssumed tht the end product of ll these industries work is finl consumer demnd. Pge 8 of 5

39 Teching nd Lerning Guide : Mtrices This cn then e linked to input-output nlysis: it is further ssumed tht chnges in demnd for ny of the finl outputs from ny industry ffect the outputs tht it requires from other industries.. Presenting the concept of input-output nlysis ij ij j Students will need sound nd forml eplntion to which they cn relte when ttempting pplied questions on their own. For emple, students could e shown tht we cn rticulte input-output nlysis s follows: Let: j e the totl output of industry j, ij e the output of industry i to industry j nd ij is constnt. If there re n industries under considertion: i n j ij y i Where y i is the output of industry i to finl demnd. Students should note tht this eqution holds since totl output of n industry must go either s inputs to other industries or to finl demnd. Comining these two equtions students cn see tht we otin: i n j ij j y i i n j ij j y i Epnding out the summtion sign nd consider for ll vlues of i from to n: Pge 9 of 5

40 Teching nd Lerning Guide : Mtrices n.. n n n n nn n n n y y y n Clerly it is possile to group the s, s etc. together in mtri form s follows: n n n n n nn n y y y n If we crete mtri of input-output coefficients, : n n n n n nn Then the mtri on the left is: I Where I is the identity mtri of order n. Students might wnt to revisit their erlier notes nd mteril t this point. Lecturers could highlight tht we often cll the column of totl outputs X, nd the column vector of outputs to finl demnd Y, then: (I )XY Pge of 5

41 Teching nd Lerning Guide : Mtrices nd hence: X(I-) - Y (ssuming tht the inverse of (I ) eists. This is n importnt result since if we re given the required outputs to finl demnd of ech industry nd provided we know the input-output coefficients we cn otin the required totl outputs of ech industry.. Delivering the concept of input-output nlysis to smll or lrger groups Worked emple Working ntionl input-output models del with mny industries. However the sic principle cn e illustrted with the id of smll scle (n) emple: / 8 / / / / 6 / 6 / / / Y 7 55 Given nd Y we cn find the output of ech of the three industries s follows: X (I-) - Y In order to find the inverse of we must first find I-: I / 8 / / / / 6 / 6 / / / 7 / 8 / / / 5 / 6 / 6 / / / nd then find the determinnt of the resultnt mtri: Epnding y column : Pge of 5

42 Teching nd Lerning Guide : Mtrices 7 / 8 5 / 6 / 6 / / ( / ) / / 6 / / ( / ) / 5 / 6 / / ( 5 ) ( ) ( 5 ) ( 7 ) ( 7 ) ( 7 ) 9 / 96 7 / 8 7 / 96 7 / 96 7 / The finding the nine cofctors: (5 / / ) 7 / ( / / ) (/ ( / 8 /6) 7 /6 (/ /6) ( 7 / / 8) (/ 5 / ) 5 / ) ( 7 / 8 /) (5 / 8 / 6) 7 / 7 / 9 / / 7 / / 8 7 / 8 9 /6 Putting these in-order forms the djugte mtri: Then tking the trnspose of the djugte mtri to form the djoint mtri: The inverse is therefore: Pge of 5

43 Teching nd Lerning Guide : Mtrices ( I ) / / 9 7 / 9 Multiplying this y the column vector of finl demnd we find pproimtely: 86 / 7 7 / / 59 / 9 7 / / / Which gives directly the outputs given (or required) from ech industry s eing 697, 5 nd 6577 respectively. Note tht we cn lso clculte the vector X-Y which gives directly tht output of ech industry tht goes to other industries rther thn s finl demnd. Worked emple: Etension Mteril Finlly, from the eqution: X (I-) - Y We cn find the mrginl effect of hving to produce one etr unit of finl demnd: X (I-) - Y Where X nd Y re the respective smll increses. Hence, if in the emple ove we needed to produce one etr unit for industry the Y vector would e: nd therefore: Y Pge of 5

44 Teching nd Lerning Guide : Mtrices X ( I ) Y / 59 / 9 7 / 9 86 / 7 86 / 7 7 / 7 / Links with the online question nk Questions focused on mtrices cn e found t Discussion Questions Students could e sked to find rel world economic stories from qulity news sources e.g. The Economist to show how input-output nlysis could e pplied. For emple, wht knockon might the demise of the Rover cr compny hve on other industries nd the UK economy? 5. ctivities Tsk One Lerning Ojectives LO: Students lern how to pply their knowledge of mtrices to input-output nlysis LO: Students lern how to drw inferences from nswers derived using mtri lger Tsk One sk students to review the worked emple ove (See.). Working in pirs, students should crete Powerpoint presenttion which summrises: - wht the worked emple ws trying to solve - the key steps to solving the input-output prolem; nd - ny prolems nd difficulties they encountered when working through the worked emple. Tsk Two Working in pirs. crete your own vlues of nd Y nd ssume n. Pge of 5

45 Teching nd Lerning Guide : Mtrices 6. Top Tips Students will wnt to prctise questions hving first ensured they hve mstered ech of the steps (See ove). Lecturers will proly wnt to crete opportunities for students to Ecel to pply their knowledge nd understnding. Higher ility students could e sked to develop their own prolem sets nd nswers. 7. Conclusion This is very much summtive or plenry topic. Crete chnces for students to work through prolem sets nd idelly offer one-to-one support s they design nd crete their own independent input-output prolem set (See 5. ove). uthored nd Produced y the METL Project Consortium 7 Pge 5 of 5