Elementary Linear Algebra

Transcription

1 Elemetry Lier Alger Ato & Rorres, th Editio Lecture Set Chpter : Systems of Lier Equtios & Mtrices

2 Chpter Cotets Itroductio to System of Lier Equtios Gussi Elimitio Mtrices d Mtri Opertios Iverses; Rules of Mtri Arithmetic Elemetry Mtrices d Method for Fidig A - Further Results o Systems of Equtios d Ivertiility Digol, Trigulr, d Symmetric Mtrices

3 - Lier Equtios Ay stright lie i y-ple c e represeted lgericlly y equtio of the form: + y = Geerl form: Defie lier equtio i the vriles,,, : = where,,, d re rel costts. The vriles i lier equtio re sometimes clled ukows.

4 - Emple (Lier Equtios) The equtios y 7, y z, d 4 7 re lier A lier equtio does ot ivolve y products or roots of vriles All vriles occur oly to the first power d do ot pper s rgumets for trigoometric, logrithmic, or epoetil fuctios. The equtios y, yzz4, d y si re ot lier A solutio of lier equtio is sequece of umers s, s,, s such tht the equtio is stisfied. The set of ll solutios of the equtio is clled its solutio set or geerl solutio of the equtio. 4

5 - Emple (Lier Equtios) Fid the solutio of = Solutio: We c ssig ritrry vlues to y two vriles d solve for the third vrile For emple = + 4s 7t, = s, = t where s, t re ritrry vlues

6 - Lier Systems m m m m A fiite set of lier equtios i the vriles,,, is clled system of lier equtios or lier system. A sequece of umers s, s,, s is clled solutio of the system A system hs o solutio is sid to e icosistet. If there is t lest oe solutio of the system, it is clled cosistet. Every system of lier equtios hs either o solutios, ectly oe solutio, or ifiitely my solutios 6

7 - Lier Systems A geerl system of two lier equtios: + y = c (, ot oth zero) + y = c (, ot oth zero) Two lie my e prllel o solutio Two lie my e itersect t oly oe poit oe solutio Two lie my coicide ifiitely my solutios 7

8 8 - Augmeted Mtrices The loctio of the +s, the s, d the =s c e revited y writig oly the rectgulr rry of umers. This is clled the ugmeted mtri for the system. It must e writte i the sme order i ech equtio s the ukows d the costts must e o the right m m m m m m m m th colum th row

9 9 - Elemetry Row Opertios The sic method for solvig system of lier equtios is to replce the give system y ew system tht hs the sme solutio set ut which is esier to solve. Sice the rows of ugmeted mtri correspod to the equtios i the ssocited system, ew systems is geerlly otied i series of steps y pplyig the followig three types of opertios to elimite ukows systemticlly. m m m m m m m m

10 - Elemetry Row Opertios Elemetry row opertios Multiply row through y ozero costt Iterchge two rows Add costt times oe row to other Algeric opertios Multiply equtio through y ozero costt Iterchge two equtios Add costt times oe equtio to other

11 - Emple (Usig Elemetry Row Opertios) z y z y z y z y z y z y z y z y z y z y z y z y R -R R -R R / z z y z y z z y z y z y 7 7 z z y z 7 7 R -R -R R -R

12 Chpter Cotets Itroductio to System of Lier Equtios Gussi Elimitio Mtrices d Mtri Opertios Iverses; Rules of Mtri Arithmetic Elemetry Mtrices d Method for Fidig A - Further Results o Systems of Equtios d Ivertiility Digol, Trigulr, d Symmetric Mtrices // Elemetry Lier Algorithm

13 - Echelo Forms A mtri is i reduced row-echelo form If row does ot cosist etirely of zeros, the the first ozero umer i the row is. We cll this ledig. If there re y rows tht cosist etirely of zeros, the they re grouped together t the ottom of the mtri. I y two successive rows tht do ot cosist etirely of zeros, the ledig i the lower row occurs frther to the right th the ledig i the higher row. Ech colum tht cotis ledig hs zeros everywhere else. A mtri tht hs the first three properties is sid to e i rowechelo form. Note: A mtri i reduced row-echelo form is of ecessity i rowechelo form, ut ot coversely.

14 4 - Emple Reduce row-echelo form: Row-echelo form:,,, 7 4 6,, 6 7 4

15 - Emple Mtrices i row-echelo form (y rel umers sustituted for the s. ) : Mtrices i reduced row-echelo form (y rel umers sustituted for the s. ) :,,,,,,

16 - Emple : ledig vrile, free vrile Solutios of lier systems

17 - Elimitio Methods A step-y-step elimitio procedure tht c e used to reduce y mtri to reduced row-echelo form System of lier equtios

18 8 - Elimitio Methods Step. Locte the leftmost colum tht does ot cosist etirely of zeros. Step. Iterchge the top row with other row, to rig ozero etry to top of the colum foud i Step Leftmost ozero colum The th d th rows i the precedig mtri were iterchged

19 - Elimitio Methods Step. If the etry tht is ow t the top of the colum foud i Step is, multiply the first row y / i order to itroduce ledig From step: 4 4 The st row of the precedig mtri ws multiplied y /. Step4. Add suitle multiples of the top row to the rows elow so tht ll etries elow the ledig ecome zeros times the st row of the precedig mtri ws dded to the rd row. 9

20 - Elimitio Methods Step. Now cover the top row i the mtri d egi gi with Step pplied to the sumtri tht remis. Cotiue i this wy util the etire mtri is i row-echelo form Leftmost ozero colum i the sumtri The st row i the sumtri ws multiplied y -/ to itroduce ledig.

21 - Elimitio Methods The lst mtri is i row-echelo form times the st row of the sumtri ws dded to the d row of the sumtri to itroduce zero elow the ledig The top row i the sumtri ws covered, d we retured gi Step. The first (d oly) row i the ew sumetri ws multiplied y to itroduce ledig. Leftmost ozero colum i the ew sumtri

22 - Elimitio Methods Step 6. Begiig with the lst ozero row d workig upwrd, dd suitle multiples of ech row to the rows ove to itroduce zeros ove the ledig s / times the rd row of the precedig mtri ws dded to the secod row times the rd row ws dded to the first row. 7 times the d row ws dded to the first row.

23 - Elimitio Methods Step~Step: the ove procedure produces row-echelo form d is clled Gussi elimitio Step~Step6: the ove procedure produces reduced row-echelo form d is clled Gussi-Jord elimitio Every mtri hs uique reduced row-echelo form ut rowechelo form of give mtri is ot uique Bck-Sustitutio To solve system of lier equtios y usig Gussi elimitio to rig the ugmeted mtri ito row-echelo form without cotiuig ll the wy to the reduced row-echelo form. Whe this is doe, the correspodig system of equtios c e solved y solved y techique clled ck-sustitutio

24 - Emple 4 Solve y Guss-Jord elimitio

25 - Emple From the computtios i emple 4, row-echelo form of the ugmeted mtri is give. To solve the system of equtios: / / 6 6 4

26 - Emple 6 Solve the system of equtios y Gussi elimitio d ck-sustitutio. y z 9 4y z 6y z 6

27 - Homogeeous Lier Systems A system of lier equtios is sid to e homogeeous if the costt terms re ll zero.... Every homogeeous system of lier equtio is cosistet, sice ll such system hve =, =,, = s solutio. This solutio is clled the trivil solutio. If there re other solutios, they re clled otrivil solutios. There re oly two possiilities for its solutios: m m There is oly the trivil solutio There re ifiitely my solutios i dditio to the trivil solutio m 7

28 8 - Emple 7 Solve the homogeeous system of lier equtios y Guss-Jord elimitio The ugmeted mtri Reducig this mtri to reduced row-echelo form The geerl solutio is Note: the trivil solutio is otied whe s = t = 4 4 t t s t s 4,,,

29 - Emple 7 (Guss-Jord Elimitio) Two importt poits: Noe of the three row opertios lters the fil colum of zeros, so the system of equtios correspodig to the reduced row-echelo form of the ugmeted mtri must lso e homogeeous system. If the give homogeeous system hs m equtios i ukows with m <, d there re r ozero rows i reduced row-echelo form of the ugmeted mtri, we will hve r <. It will hve the form: k k (Theorem..) kr () () () k k kr () () () 9

30 Theorem..,.. Theorem.. Free Vrile Theorem for Homogeeous Systems If homogeeous lier system hs ukows, d if the reduced row echelo form of its ugmeted mtri hs r ozero rows, the the system hs -r free vriles. Theorem.. A homogeeous lier system with more ukows th equtios hs ifiitely my solutios. Remrk This theorem pplies oly to homogeeous system! A ohomogeeous system with more ukows th equtios eed ot e cosistet; however, if the system is cosistet, it will hve ifiitely my solutios. e.g., two prllel ples i -spce

31 Lier Systems i Three Ukows

32 Chpter Cotets Itroductio to System of Lier Equtios Gussi Elimitio Mtrices d Mtri Opertios Iverses; Rules of Mtri Arithmetic Elemetry Mtrices d Method for Fidig A - Further Results o Systems of Equtios d Ivertiility Digol, Trigulr, d Symmetric Mtrices // Elemetry Lier Algorithm

33 - Defiitio d Nottio A mtri is rectgulr rry of umers. The umers i the rry re clled the etries i the mtri A geerl m mtri A is deoted s A m The etry tht occurs i row i d colum j of mtri A will e deoted ij or A ij. If ij is rel umer, it is commo to e referred s sclrs The precedig mtri c e writte s [ ij ] m or [ ij ] m m

34 - Defiitio Two mtrices re defied to e equl if they hve the sme size d their correspodig etries re equl If A = [ ij ] d B = [ ij ] hve the sme size, the A = B if d oly if ij = ij for ll i d j If A d B re mtrices of the sme size, the the sum A + B is the mtri otied y ddig the etries of B to the correspodig etries of A. 4

35 - Defiitio The differece A B is the mtri otied y sutrctig the etries of B from the correspodig etries of A If A is y mtri d c is y sclr, the the product ca is the mtri otied y multiplyig ech etry of the mtri A y c. The mtri ca is sid to e the sclr multiple of A If A = [ ij ], the ca ij = ca ij = c ij

36 6 - Defiitios If A is mr mtri d B is r mtri, the the product AB is the m mtri whose etries re determied s follows. (AB) m = A mr B r AB ij = i j + i j + i j + + ir rj r rj r r j j mr m m ir i i r r AB

37 - Emple Multiplyig mtrices A 6 4 B

38 - Emple 6 Determie whether product is defied Mtrices A: 4, B: 4 7, C: 7 8

39 - Prtitioed Mtrices A mtri c e prtitioed ito smller mtrices y isertig horizotl d verticl rules etwee selected rows d colums For emple, three possile prtitios of 4 mtri A: The prtitio of A ito four 4 sumtrices A, A, A, A A A d A 4 A A 4 The prtitio of A ito its row 4 r mtrices r, r, d r A 4 r The prtitio of A ito its 4 r colum mtrices c, c, c, 4 d c 4 A c c c c

40 4 - Multiplictio y Colums d y Rows It is possile to compute prticulr row or colum of mtri product AB without computig the etire product: jth colum mtri of AB = A[jth colum mtri of B] ith row mtri of AB = [ith row mtri of A]B If,,..., m deote the row mtrices of A d,,..., deote the colum mtrices of B,the B B B B AB A A A A AB m m AB computed colum y colum AB computed row y row

41 - Emple 7 Multiplyig mtrices y rows d y colums 6 4 A B 4

42 4 - Mtri Products s Lier Comitios Let The The product A of mtri A with colum mtri is lier comitio of the colum mtrices of A with the coefficiets comig from the mtri m m m A d m m m m m m A

43 - Emple 8 4

44 - Emple 9 44

45 4 - Mtri Form of Lier System Cosider y system of m lier equtios i ukows: The mtri A is clled the coefficiet mtri of the system The ugmeted mtri of the system is give y m m m m m m m m m m m m A m m m m A

46 - Defiitios If A is y m mtri, the the trspose of A, deoted y A T, is defied to e the m mtri tht results from iterchgig the rows d colums of A Tht is, the first colum of A T is the first row of A, the secod colum of A T is the secod row of A, d so forth If A is squre mtri, the the trce of A, deoted y tr(a), is defied to e the sum of the etries o the mi digol of A. The trce of A is udefied if A is ot squre mtri. For mtri A = [ ij ], tr( A) ii i 46

47 - Emple & Trspose: (A T ) ij = (A) ji 6 4 A Trce of mtri: B 47

48 Chpter Cotets Itroductio to System of Lier Equtios Gussi Elimitio Mtrices d Mtri Opertios Iverses; Rules of Mtri Arithmetic Elemetry Mtrices d Method for Fidig A - Further Results o Systems of Equtios d Ivertiility Digol, Trigulr, d Symmetric Mtrices // Elemetry Lier Algorithm 48

49 -4 Properties of Mtri Opertios For rel umers d,we lwys hve =, which is clled the commuttive lw for multiplictio. For mtrices, however, AB d BA eed ot e equl. Equlity c fil to hold for three resos: The product AB is defied ut BA is udefied. AB d BA re oth defied ut hve differet sizes. It is possile to hve AB BA eve if oth AB d BA re defied d hve the sme size. 49

50 Theorem.4. (Properties of Mtri Arithmetic) Assumig tht the sizes of the mtrices re such tht the idicted opertios c e performed, the followig rules of mtri rithmetic re vlid: A + B = B + A (commuttive lw for dditio) A + (B + C) = (A + B) + C (ssocitive lw for dditio) A(BC) = (AB)C (ssocitive lw for multiplictio) A(B + C) = AB + AC (left distriutive lw) (B + C)A = BA + CA (right distriutive lw) A(B C) = AB AC, (B C)A = BA CA (B + C) = B + C, (B C) = B C (+)C = C + C, (-)C = C C (C) = ()C, (BC) = (B)C = B(C) Note: the ccelltio lw is ot vlid for mtri multiplictio!

51 -4 Emple

52 -4 Zero Mtrices A mtri, ll of whose etries re zero, is clled zero mtri A zero mtri will e deoted y If it is importt to emphsize the size, we shll write m for the m zero mtri. I keepig with our covetio of usig oldfce symols for mtrices with oe colum, we will deote zero mtri with oe colum y

53 -4 Emple The ccelltio lw does ot hold AB=AC = A 4 B 4 C 7 D 8 6 4

54 Theorem.4. (Properties of Zero Mtrices) Assumig tht the sizes of the mtrices re such tht the idicted opertios c e performed,the followig rules of mtri rithmetic re vlid A + = + A = A A A = A = A A = ; A = If ca=, the c= or A= 4

55 -4 Idetity Mtrices A squre mtri with s o the mi digol d s off the mi digol is clled idetity mtri d is deoted y I, or I for the idetity mtri If A is m mtri, the AI = A d I m A = A Emple 4 A idetity mtri plys the sme role i mtri rithmetic s the umer plys i the umericl reltioships = =

56 Theorem.4. If R is the reduced row-echelo form of mtri A, the either R hs row of zeros or R is the idetity mtri I 6

57 -4 Ivertile If A is squre mtri, d if mtri B of the sme size c e foud such tht AB = BA = I, the A is sid to e ivertile d B is clled iverse of A. If o such mtri B c e foud, the A is sid to e sigulr. Remrk: The iverse of A is deoted s A - Not every (squre) mtri hs iverse A iverse mtri hs ectly oe iverse 7

58 -4 Emple & 6 Verify the iverse requiremets A mtri with o iverse is sigulr A B 6 4 A 8

59 -4 Theorems Theorem.4.4 If B d C re oth iverses of the mtri A, the B = C Theorem.4. The mtri A c d is ivertile if d c, i which cse the iverse is give y the formul d A d c c 9

60 Theorem.4.6 If A d B re ivertile mtrices of the sme size,the AB is ivertile d (AB) - = B - A - Emple 7 A B AB

61 -4 Powers of Mtri If A is squre mtri, the we defie the oegtive iteger powers of A to e A I A AA A ( ) fctors If A is ivertile, the we defie the egtive iteger powers to e A ( A ) A A A ( ) fctors Theorem.4.7 (Lws of Epoets) If A is squre mtri d r d s re itegers, the A r A s = A r+s, (A r ) s = A rs 6

62 Theorem.4.8 (Lws of Epoets) If A is ivertile mtri, the: A - is ivertile d (A - ) - = A A is ivertile d (A ) - = (A - ) for =,,, For y ozero sclr k, the mtri ka is ivertile d (ka) - = (/k)a - 6

63 -4 Emple 8 Powers of mtri A A A =? A - =? 6

64 -4 Polyomil Epressios Ivolvig Mtrices If A is squre mtri, sy mm, d if p() = is y polyomil, the we defie p(a) = I + A + + A where I is the mm idetity mtri. Tht is, p(a) is the mm mtri tht results whe A is sustituted for i the ove equtio d is replced y I 64

65 -4 Emple 9 (Mtri Polyomil) 6

66 Theorems.4.9 (Properties of the Trspose) If the sizes of the mtrices re such tht the stted opertios c e performed, the ((A T ) T = A (A + B) T = A T + B T d (A B) T = A T B T (ka) T = ka T, where k is y sclr (AB) T = B T A T 66

67 Theorem.4. (Ivertiility of Trspose) If A is ivertile mtri, the A T is lso ivertile d (A T ) - = (A - ) T Emple A T A 67

68 Chpter Cotets Itroductio to System of Lier Equtios Gussi Elimitio Mtrices d Mtri Opertios Iverses; Rules of Mtri Arithmetic Elemetry Mtrices d Method for Fidig A - Further Results o Systems of Equtios d Ivertiility Digol, Trigulr, d Symmetric Mtrices // Elemetry Lier Algorithm 68

69 - Elemetry Row Opertio A elemetry row opertio (sometimes clled just row opertio) o mtri A is y oe of the followig three types of opertios: Iterchge of two rows of A Replcemet of row r of A y cr for some umer c Replcemet of row r of A y the sum r + cr of tht row d multiple of other row r of A 69

70 - Elemetry Mtri A elemetry mtri is mtri produced y pplyig ectly oe elemetry row opertio to I E ij is the elemetry mtri otied y iterchgig the i- th d j-th rows of I E i (c) is the elemetry mtri otied y multiplyig the i- th row of I y c E ij (c) is the elemetry mtri otied y ddig c times the j-th row to the i-th row of I, where i j 7

71 - Emple Elemetry Mtrices d Row Opertios 7

72 - Elemetry Mtrices d Row Opertios Theorem.. Suppose tht E is mm elemetry mtri produced y pplyig prticulr elemetry row opertio to I m, d tht A is m mtri. The EA is the mtri tht results from pplyig tht sme elemetry row opertio to A 7

73 - Emple (Usig Elemetry Mtrices) 7

74 - Iverse Opertios If elemetry row opertio is pplied to idetity mtri I to produce elemetry mtri E, the there is secod row opertio tht, whe pplied to E, produces I ck gi Row opertio o I Tht produces E Row opertio o E Tht produces I Multiply row i y c Multiply row i y /c Iterchge row i d j Add c times row i to row j Iterchge row i d j Add -c times row i to row j 74

75 - Iverse Opertios Emples Multiply the d y 7 Multiply the d y /7 Iterchge the st d d row Iterchge the st d d row Add times the d to the st row row Add - times the d to the st row row

76 Theorem.. Elemetry Mtrices d Nosigulrity Ech elemetry mtri is osigulr, d its iverse is itself elemetry mtri. More precisely, E - ij = E ji (= E ij ) E i (c) - = E i (/c) with c E ij (c) - = E ij (-c) with i j 76

77 Theorem..(Equivlet Sttemets) If A is mtri, the the followig sttemets re equivlet, tht is, ll true or ll flse A is ivertile A = hs oly the trivil solutio The reduced row-echelo form of A is I A is epressile s product of elemetry mtrices 77

78 - A Method for Ivertig Mtrices To fid the iverse of ivertile mtri A, we must fid sequece of elemetry row opertios tht reduces A to the idetity d the perform this sme sequece of opertios o I to oti A - Remrk Suppose we c fid elemetry mtrices E, E,, E k such tht the E k E E A = I A - = E k E E I 78

79 - Emple 4 (Usig Row Opertios to Fid A - ) Fid the iverse of Solutio: A To ccomplish this we shll djoi the idetity mtri to the right side of A, therey producig mtri of the form [A I] We shll pply row opertios to this mtri util the left side is reduced to I; these opertios will covert the right side to A -, so tht the fil mtri will hve the form [I A - ] 8 79

80 - Emple 4 8

81 - Emple 4 (cotiue) 8

82 - Emple Cosider the mtri 6 4 A 4 Apply the procedure of emple 4 to fid A - 8

83 - Emple 6 Accordig to emple 4, A is ivertile mtri. hs oly trivil solutio 8 8 A 8

84 Chpter Cotets Itroductio to System of Lier Equtios Gussi Elimitio Mtrices d Mtri Opertios Iverses; Rules of Mtri Arithmetic Elemetry Mtrices d Method for Fidig A - Further Results o Systems of Equtios d Ivertiility Digol, Trigulr, d Symmetric Mtrices // Elemetry Lier Algorithm 84

85 Theorems.6. Every system of lier equtios hs either o solutios, ectly oe solutio, or i fiitely my solutios. 8

86 Theorem.6. If A is ivertile mtri, the for ech mtri, the system of equtios A = hs ectly oe solutio, mely, = A m m m m m m m m A

87 -6 Emple 87

88 -6 Lier Systems with Commo Coefficiet Mtri To solve sequece of lier systems, A =, A =,, A k = k, with commo coefficiet mtri A If A is ivertile, the the solutios = A -, = A -,, k = A - k A more efficiet method is to form the mtri [A k ] By reducig it to reduced row-echelo form we c solve ll k systems t oce y Guss-Jord elimitio. 88

89 -6 Emple Solve the system

90 Theorems.6. Let A e squre mtri If B is squre mtri stisfyig BA = I, the B = A - If B is squre mtri stisfyig AB = I, the B = A - 9

91 Theorem.6.4 (Equivlet Sttemets) If A is mtri, the the followig sttemets re equivlet A is ivertile A = hs oly the trivil solutio The reduced row-echelo form of A is I A is epressile s product of elemetry mtrices A = is cosistet for every mtri A = hs ectly oe solutio for every mtri 9

92 Theorem.6. Let A d B e squre mtrices of the sme size. If AB is ivertile, the A d B must lso e ivertile. Let A e fied m mtri. Fid ll m mtrices such tht the system of equtios A= is cosistet. 9

93 -6 Emple Fid,, d such tht the system of equtios is cosistet. 9

94 -6 Emple 4 Fid,, d such tht the system of equtios is cosistet. 94 8

95 Chpter Cotets Itroductio to System of Lier Equtios Gussi Elimitio Mtrices d Mtri Opertios Iverses; Rules of Mtri Arithmetic Elemetry Mtrices d Method for Fidig A - Further Results o Systems of Equtios d Ivertiility Digol, Trigulr, d Symmetric Mtrices // Elemetry Lier Algorithm 9

96 -7 Digol Mtri A squre mtri A is m with m = ; the (i,j)-etries for i m form the mi digol of A A digol mtri is squre mtri ll of whose etries ot o the mi digol equl zero. By dig(d,, d m ) is met the mm digol mtri whose (i,i)-etry equls d i for i m 96

97 97-7 Properties of Digol Mtrices A geerl digol mtri D c e writte s A digol mtri is ivertile if d oly if ll of its digol etries re ozero Powers of digol mtrices re esy to compute d d d D d d d D / / / k k k k d d d D

98 -7 Properties of Digol Mtrices Mtri products tht ivolve digol fctors re especilly esy to compute 98

99 -7 Trigulr Mtrices A m lower-trigulr mtri L stisfies (L) ij = if i < j, for i m d j A m upper-trigulr mtri U stisfies (U) ij = if i > j, for i m d j A uit-lower (or upper)-trigulr mtri T is lower (or upper)-trigulr mtri stisfyig (T) ii = for i mi(m,) 99

100 -7 Emple (Trigulr Mtrices) A geerl 44 upper trigulr mtri A geerl 44 lower trigulr mtri The trigulr mtri oth upper trigulr d lower trigulr A squre mtri i row-echelo form is upper trigulr

101 Theorem.7. The trspose of lower trigulr mtri is upper trigulr, d the trspose of upper trigulr mtri is lower trigulr The product of lower trigulr mtrices is lower trigulr, d the product of upper trigulr mtrices is upper trigulr A trigulr mtri is ivertile if d oly if its digol etries re ll ozero The iverse of ivertile lower trigulr mtri is lower trigulr, d the iverse of ivertile upper trigulr mtri is upper trigulr

102 -7 Emple Cosider the upper trigulr mtrices 4 A B

103 -7 Symmetric Mtrices A (squre) mtri A for which A T = A, so tht A ij = A ji for ll i d j, is sid to e symmetric. Emple d d d d

104 Theorem.7. If A d B re symmetric mtrices with the sme size, d if k is y sclr, the A T is symmetric A + B d A B re symmetric ka is symmetric Remrk The product of two symmetric mtrices is symmetric if d oly if the mtrices commute, i.e., AB = BA Emple 4 4 4

105 Theorem.7. If A is ivertile symmetric mtri, the A - is symmetric. Remrk: I geerl, symmetric mtri eeds ot e ivertile. The products AA T d A T A re lwys symmetric

106 -7 Emple 6 6

107 Theorem.7.4 If A is ivertile mtri, the AA T d A T A re lso ivertile 7