Algebra Of Matrices & Determinants

Transcription

1 lgebr Of Mtrices & Determinnts Importnt erms Definitions & Formule 0 Mtrix - bsic introduction: mtrix hving m rows nd n columns is clled mtrix of order m n (red s m b n mtrix) nd mtrix of order lso in generl mens n element ling in e m n is depicted s i row nd No of elements in e mtrix is given s ( m)( n ) m n 02 pes of Mtrices: ) Column mtrix: mtrix hving onl one column is clled column mtrix or column vector Generl nottion: m j column ; i j N m n b) Row mtrix: mtrix hving onl one row is clled row mtrix or row vector Generl nottion: n c) Squre mtrix: It is mtrix in which e number of rows is equl to e number of columns ie n m n mtrix is sid to constitute squre mtrix if m n nd is known s squre mtrix of order n is squre mtrix of order 0 2 Generl nottion: d) Digonl mtrix: squre mtrix ie ll its non- digonl elements re zero nn is sid to be digonl mtrix if 0 when i j mm is digonl mtrix of order lso ere is one more nottion specificll used for e digonl mtrices For instnce consider e dig mtrix depicted bove it cn be lso written s Note t e elements 22 mm of squre mtrix of order m re sid to m n constitute e principl digonl or simpl e digonl of e squre mtrix nd ese elements re known s digonl elements of mtrix e) Sclr mtrix: digonl mtrix equl ie is sid to be sclr mtrix if its digonl elements re mm 0 when i j k when i j for some constnt k is sclr mtrix of order f) Unit or Identit mtrix: squre mtrix if i j 0 if i j List Of Formule for Clss XII B Mohmmed bbs (II PCMB '') is sid to be n identit mtrix if mn List Of Formule B Mohmmed bbs Pge - [0]

2 unit mtrix cn lso be defined s e sclr mtrix ech of whose digonl elements is unit We denote e identit mtrix of order m b I m or I 0 0 I I 0 g) Zero mtrix or Null mtrix: mtrix is sid to be zero mtrix or null mtrix if ech of its elements is zero h) Horizontl mtrix: m n mtrix is sid to be horizontl mtrix if m < n j) ringulr mtrix: Lower tringulr mtrix: squre mtrix is clled lower tringulr mtrix if 0 when i j List Of Formule for Clss XII B Mohmmed bbs (II PCMB '') i) Verticl mtrix: m n mtrix is sid to be vector mtrix if m > n Upper tringulr mtrix: squre mtrix is clled n upper tringulr mtrix if 0 when i j Properties of mtrix ddition: Commuttive propert: B B ssocitive propert: B C B C Cncelltion lws: i) Left cncelltion- B C B C ii) Right cncelltion- B C B C 04 Properties of mtrix multipliction: Note t e product B is defined onl when e number of columns in mtrix is equl to e number of rows in mtrix B If nd B re m n nd n p mtrices respectivel en mtrix B will be n m p mtrix mtrix ie order of mtrix B will be m p In e product B is clled e pre-fctor nd B is clled e post-fctor If e product B is possible en it is not necessr t e product B is lso possible If is m n mtrix nd bo B nd B re defined en B will be n m mtrix If is n n mtrix nd I n be e unit mtrix of order n en In In Mtrix multipliction is ssocitive ie BC BC Mtrix multipliction is distributive over e ddition ie B C B C Idempotent mtrix: squre mtrix is sid to be n idempotent mtrix if 0 0 For exmple rnspose of Mtrix: If be n m n mtrix en e mtrix obtined b interchnging m n e rows nd columns of mtrix is sid to be trnspose of mtrix he trnspose of is denoted b c or or ie if en mn ji nm 2 0 For exmple List Of Formule B Mohmmed bbs Pge - [02]

3 List Of Formule for Clss XII B Mohmmed bbs (II PCMB '') Properties of rnspose of mtrices: B B B B k k where k is n constnt B B BC C B 06 Smmetric mtrix: squre mtrix is sid to be smmetric mtrix if ht is if en ji For exmple: h g 2 i h b f 2 2i g f c 2i 4 07 Skew-smmetric mtrix: squre mtrix is sid to be skew-smmetric mtrix if ie if en ji 0 For exmple: Fcts ou should know: Note t ji ii ii 2 ii 0 {Replcing j b i } ht is ll e digonl elements in skew-smmetric mtrix re zero he mtrices nd re smmetric mtrices For n squre mtrix is smmetric mtrix nd is skew- smmetric mtrix lws lso n squre mtrix cn be expressed s e sum of smmetric nd skew-smmetric mtrix ie P Q where P is smmetric mtrix nd Q is 2 2 skew- smmetric mtrix 08 Orogonl mtrix: mtrix is sid to be orogonl if I where is e trnspose of 09 Invertible Mtrix: If is squre mtrix of order m nd if ere exists noer squre mtrix B of e sme order m such t B B I en B is clled e inverse mtrix of nd it is denoted b mtrix hving n inverse is sid to be invertible It is to note t if B is inverse of en is lso e inverse of B In oer words if it is known t B B I en B B 0 Determinnts Minors & Cofctors: ) Determinnt: unique number (rel or complex) cn be ssocited to ever squre mtrix of order m his number is clled e determinnt of e squre mtrix where i j element of b b For instnce if en determinnt of mtrix is written s det nd its c d c d vlue is given b d bc b) Minors: Minors of n element of determinnt (or determinnt corresponding to mtrix ) is e determinnt obtined b deleting its i row nd j column in which lies Minor of is denoted b M Hence we cn get 9 minors corresponding to e 9 elements of ird order i e determinnt List Of Formule B Mohmmed bbs Pge - [0]

4 c) Cofctors: Cofctor of n element denoted b minor of Sometimes C is used in plce of i j is defined b to denote e cofctor of element M where M is djoint of squre mtrix: Let be squre mtrix lso ssume B where is e cofctor of e elements in mtrix hen e trnspose B of mtrix B is clled e djoint of mtrix nd it is denoted b dj o find djoint of 2 2 mtrix: Follow b d b dj c d c 2 For exmple consider squre mtrix of order s 2 4 en in order to find e djoint of mtrix we find mtrix B (formed b e cofctors of elements of mtrix s mentioned bove in e definition) ie B 0 4 Hence dj B Singulr mtrix & Non-singulr mtrix: ) Singulr mtrix: squre mtrix is sid to be singulr if 0 ie its determinnt is zero b) Non-singulr mtrix: squre mtrix is sid to be non-singulr if squre mtrix is invertible if nd onl if is non-singulr 4 Elementr Opertions or rnsformtions of Mtrix: he following ree opertions pplied on e row (or column) of mtrix re clled elementr row (or column) trnsformtions ) Interchnge of n two rows (or columns): When i row (or column) of mtrix is interchnged wi e j row (or column) it is denoted s Ri R j (or Ci C j ) b) Multipling ll elements of row (or column) of mtrix b non-zero sclr: When e row (or column) of mtrix is multiplied b sclr k it is denoted s Ri kri (or Ci kci ) c) dding to e elements of row (or column) e corresponding elements of n oer row (or column) multiplied b n sclr k: When k times e elements of j row (or column) is dded to e corresponding elements of e i row (or column) it is denoted s Ri Ri krj (or C C kc ) i i j 4 Inverse or reciprocl of squre mtrix: If is squre mtrix of order n en mtrix B (if such mtrix exists) is clled e inverse of if B B lso note t e inverse of squre mtrix is denoted b nd we write B Inverse of squre mtrix exists if nd onl if is non-singulr mtrix ie 0 If B is inverse of en is lso e inverse of B 5 lgorim to find Inverse of mtrix b Elementr Opertions or rnsformtions: B Row rnsformtions: SEP- Write e given squre mtrix s List Of Formule for Clss XII B Mohmmed bbs (II PCMB '') NOE: In cse fter ppling one or more elementr row (or column) opertions on = I (or = I) if we obtin ll zeros in one or more rows of e mtrix on LHS en does not exist I n I n i List Of Formule B Mohmmed bbs Pge - [04]

5 SEP2- Perform sequence of elementr row opertions successivel on on e LHS nd prefctor fctor I n on e RHS till we obtin e result I n B (or I n B ) SEP- Write B B Column rnsformtions: SEP- Write e given squre mtrix s I n SEP2- Perform sequence of elementr column opertions successivel on on e LHS nd post post-fctor I n on e RHS till we obtin e result I n B SEP- Write B 6 lgorim to find b Determinnt meod: SEP- Find SEP2- If 0 en write is singulr mtrix nd hence not invertible Else write is non-singulr mtrix nd hence invertible SEP- Clculte e cofctors of elements of mtrix SEP4- Write e mtrix of cofctors of elements of nd en obtin its trnspose to get dj SEP5- Find e inverse of b using e reltion dj 7 Properties ssocited wi e Inverse of Mtrix & e Determinnts: ) B I B b) I or c) B B d) BC I C B e) f) g) dj dj I h) dj B djbdj i) dj ( dj ) j) dj dj k) n dj if 0 where n is order of dj l) B B m) n) provided mtrix is invertible o) k k n where n is order of squre mtrix nd k is n sclr If is non-singulr mtrix of order n en If is non-singulr mtrix of order n en n where n is order of n dj [ point ( k) given bove] n2 dj dj 8 Properties of Determinnts: ) If n two rows or columns of determinnt re proportionl or identicl en its vlue is zero b c 2 b2 c2 0 [s R nd R re e sme b c b) he vlue of determinnt remins unchnged if its rows nd columns re interchnged b c List Of Formule for Clss XII B Mohmmed bbs (II PCMB '') 2 b c b b b b c c c c 2 Here rows nd columns hve been interchnged but ere is no effect on e vlue of determinnt c) If ech element of row or column of determinnt is multiplied b constnt k en e vlue of new determinnt is k times e vlue of e originl determinnt List Of Formule B Mohmmed bbs Pge - [05]

6 b c 2 b2 c2 2 b2 c2 k 2 b2 c2 k b c b c b c k kb kc b c d) If n two rows or columns re interchnged en e determinnt retins its bsolute vlue but its sign is chnged b c b c 2 b2 c2 2 b2 c2 [Here R R b c b c e) If ever element of some column or row is e sum of two terms en e determinnt is equl to e sum of two determinnts; one contining onl e first term in plce of ech sum e oer onl e second term he remining elements of bo determinnts re e sme s given in e originl determinnt b c b c b c b c b c b c b c b c b c 9 re of tringle: re of tringle whose vertices re x x nd x x2 2 sq units 2 x 2 2 x is given b s e re is positive quntit we tke bsolute vlue of e determinnt given bove 2 2 x re colliner en 0 he eqution of line pssing rough e points x nd x2 2 cn be obtined b e expression given here: x If e points x x nd x x Solutions of Sstem of Liner equtions: ) Consistent nd Inconsistent sstem: sstem of equtions is consistent if it hs one or more solutions oerwise it is sid to be n inconsistent sstem In oer words n inconsistent sstem of equtions hs no solution b) Homogeneous nd Non-homogeneous sstem: sstem of equtions X B is sid to be homogeneous sstem if B 0 Oerwise it is clled non-homogeneous sstem of equtions 2 Solving of sstem of equtions b Mtrix meod Inverse Meod : erefore Consider e following sstem of equtions x b c z d SEP- ssume x b c z d x b c z d b c b c b c d B d nd 2 X d SEP2- Find Now ere m be following situtions: ) List Of Formule for Clss XII B Mohmmed bbs (II PCMB '') x z 0 exists It implies t e given sstem of equtions is consistent nd erefore e sstem hs unique solution In t cse write X B List Of Formule B Mohmmed bbs Pge - [06]

7 X B where dj hen b using e definition of equlit of mtrices we cn get e vlues of x nd z b) 0 does not exist It implies t e given sstem of equtions m be consistent or inconsistent In order to check proceed s follow: Find dj B Now we m hve eier dj B 0 or dj B 0 If dj B 0 en e given sstem m be consistent or inconsistent o check put z k in e given equtions nd proceed in e sme mnner in e new two vribles sstem sstem of equtions ssuming d c k i s constnt nd if dj B 0 List Of Formule for Clss XII B Mohmmed bbs (II PCMB '') i i en e given sstem is inconsistent wi no solutions List Of Formule B Mohmmed bbs Pge - [07]