Matrix. Definition 1... a1 ... (i) where a. are real numbers. for i 1, 2,, m and j = 1, 2,, n (iii) A is called a square matrix if m n.

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1 Mtrx Defto () s lled order of m mtrx, umer of rows ( 橫行 ) umer of olums ( 直列 ) m m m where j re rel umers () B j j for,,, m d j =,,, () s lled squre mtrx f m (v) s lled zero mtrx f (v) s lled detty mtrx f () s squre mtrx j for j () j for j Property (ddto, sutrto d slr multplto) d B re two mtres of order m d k s slr () B j j m m k k () k kj km Property () () ( B C) ( B) C B B () k( B) k kb x y z w Exerse 4 Gve tht, fd the vlues of x, y, w d z [,,, ] x y z w 5 Mtrx

2 Property 5 (Mtrx multplto d rspose) () Let M m p d B M p d j =,,,, the B M m d Bj p k k kj for,,, m () Gve M m p, the M pm d j j Property 6 () ( C) E ( CE) () ( B) C C BC () ( C D) C D (v) (v) B B (v) where s rel slr (v) ( C) C (v) I geerl, B B e B B B, B ( B)( B) (x) B does t mply or B (x) B does t mply B I 4 Exerse 7 Let, B, C, D d E Evlute () ( B) C 4 9 () ( B C) B 57 6 () (C) 9 (d) ( ) B D EE 8 7 Exerse 8 () Fd mtrx suh tht 7 7 () If x, fd the vlues of x d [, ] x Mtrx

3 Mtrx Exerse 9 Let 4 () Evlute I 5 4 () Hee, or otherwse, evlute () Hee, or otherwse, fd mtrx B suh tht I B B 8 6 Exerse Let () Fd d, () Usg () or otherwse, evlute Exerse Let, B where () Prove tht for ll postve teger, ) ( () Hee, or otherwse, evlute for y postve teger, () B) ( ) )( ( () B ) ( ] ) )[( ( ) ( Defto () s lled symmetr f () s lled skew-symmetr f

4 Defto (Determts) () d d () d e f = e + fg + dh eg d fh g h or you use the method of oftor: Property 4 () () If two rows or olums re proportol, the, e k k k g h () Let B e mtrx oted from y terhgg y rows (olums), the det( B) det( ), e d g e h f g d h e f (v) sd g r rse rh sf rs d g e h f (v) d g k e kd h kg f d g e h f or d k g e k h f k d g e h f (v) det( B) det( )det( B) (v) det( ) det( ) Exerse 5 x () If 8 x, fd the vlues of x [, 6] () Ftorze () [ ] () x y x y y x y x x y x y [ ( x y)( xy x y )] Mtrx 4

5 Exerse 6 Prove tht ( )( )( ) Defto 7 (Iverse) () squre mtrx of order s sd to e o sgulr or vertle ff there exsts squre mtrx B suh tht B B I he mtrx B s lled the verse of d deoted y () squre mtrx s sgulr or NO vertle ff does t exst Property 8 () Iverse of vertle mtrx s uque () If det( ), the s o sgulr d dj ( ) trspose of the oftor of, where dj () s the djot of whh s the heorem 9 () squre mtrx s o sgulr ff () squre mtrx s sgulr ff Exerse Property Fd the verse of Gve s squre d vertle () If B, the B () If B C, the B C () s lso vertle d If d B re o-sgulr, the ( Exerse Let B ) B 5 () Evlute 8 4I () Hee, or otherwse, fd (6 mrks) [96] 8 4 Exerse Let d P 5 () Fd P P () Fd, where s postve teger (6 mrks) [94] Mtrx 5

6 4 Exerse 4 Let, B, C B B d let e postve teger 4 () Fd C d C, () Show tht C pc qi for some tegers p d q Exerse 5 () For y dgol mtrx D,, R Prove tht D for ll postve teger () If, fd mtrx B of the form so tht B B s dgol mtrx 4 h k / () Hee, express, where s postve teger, the form of mtrx 4 / Exerse 6 () Show tht f s mtrx suh tht, the det = 74 () Gve tht B 67, use (), or otherwse, to show det B I Hee dedue tht det B 4 I (7 mrks) [9] Exerse 7 Let P e o sgulr rel mtrx d Q, where d re two dstt rel umers Defe M P QP d deote the detty mtrx y I () Fd rel umers d, terms of d, suh tht M () Prove tht det( M I M I) ( ) (6 mrks) [7] Mtrx 6

7 Mtrx 7 Exerse 8 () 8, ()() () Exerse 9 () () p, q (d)

8 Exerse ()() 6 I, () -, I () or 8 Exerse () () (d), Mtrx 8

9 Exerse ()() p, q (v) [84] he mtrx stsfes the odto () polyoml f ( x) s defed y f ( x) det xi x x x Wrte dow the polyoml f ( x) wth oeffets expressed terms of, d Evlute the mtrx f( ) I () Usg (), or otherwse, express the form I, where d re rel umers Hee fd 9 for [86] Let M e the set of rel mtres For y U M, let det U d U deote respetvely the determt d trspose of U Let F { U M : U U d detu } () () If U F, show tht detu () Show tht U F f d oly f U for some rel, suh tht w x () Let B e M y z p () Show tht f there exst U F d p, q R suh tht UBU, q the B B, e, x y () Show tht f s B B, the there exstsu F suh tht UBU t for some s, t R Mtrx 9

10 [87] Let M e the set of rel mtres d I e the detty mtrx of order () For y M, show tht f = I, the det = () Let B M suh tht B + B + I = () Show tht B = I d B ( B I ) () Smplfy I B B B () If B, show tht d d () Fd mtrx M M wth tegrl etres suh tht M I d M I [89] Let, B () Fd B d B, where B deotes the trspose of B () For eh of the mtres B d B verse f t exsts, determe whether t s vertle, d fd ts [5 mrks] [9] () Let X d Y e two squre mtres suh tht XY YX () X Y X XY Y, () X Y r C r X r Y r for =, 4, 5, Prove tht (Note : For y squre mtrx, defe I ) [ mrks] () By usg ()() d osderg 4, or otherwse, fd 4 [4 mrks] () If X d Y re squre mtres, () prove tht X Y X XY Y mples XY YX ; () prove tht X Y X X Y XY Y does NO mply XY YX (Ht : Cosder prtulr X d Y, eg X,Y ) [8 mrks] Mtrx

11 [9] Ftorze the determt [4 mrks] os s [9] () Let Prove y mthemtl duto tht s os os s s os for =,, () Let M :, R d e postve teger () For y X, Y M, show tht (I) XY M, (II) XY YX, (III) If X, the X exsts d X M () For y X M, show tht there exst r d R suh tht os s X r s os Hee fd ll X M suh tht X () If Y, B M d Y B, show tht there exsts X M suh tht X d Y BX Hee fd ll Y M suh tht Y [ mrks] [ mrks] [9] () Show tht f s mtrx suh tht, the det = () 74 Gve tht B 67, use (), or otherwse, to show det B I Hee dedue tht det B 4 I [7 mrks] 8 4 [94] Let d P 5 () Fd P P () Fd, where s postve teger [6 mrks] Mtrx

12 [94] () If, d re the roots of x pxq, fd u equto whose roots re (), d Solve the equto x x x Hee, or otherwse, solve the equto x 8x 6x 9 [6 mrks] [95] () Let Prove tht, where, R d for ll postve tegers () Hee, or otherwse, evlute 95 [6 mrks] [96] Let 5 () Evlute 8 4I () Hee, or otherwse, fd [6 mrks] [96] () Solve the equto det (*) [ mrks] () Let, ( < ) e the roots of (*) Fd two o-zero vetors (x, y ) d (x, y ) suh tht =, Show tht (x, y ) d (x, y ) re lerly depedet Let x P y x y Fd x y x y P d evlute P P [9 mrks] () Evlute 996 [ mrks] Mtrx

13 [97] () Let e o-sgulr mtrx Show tht x det xi det x I () Let = det () Show tht 4 s root of det xi () Solve det xi d hee fd the other roots surd form [7 mrks] os s [97] () () Let S = : R s os Show tht for y mtres d B S, B s lso S () Let () = Prove tht os s s os for y postve teger [4 mrks] () Let M = where, R d () Show tht M = k() for some rel umers k d Express k, os d s terms of d () If, prove tht there exsts postve teger suh tht M s dgol p (e of the form ) f d oly f q t s rtol () If =, fd ll postve tegers suh tht M s dgol [ mrks] [98] Let where,,, d R, d det = d () Show tht for some k R ( mrks) k k () Fd P the form of suh tht r P for some, R s If + d =, fd Q the form of suh tht PP Q for some R (5 mrks) 7 () Fd S suh tht S S for some R 6 4 Hee, or otherwse, evlute where s postve teger (7 mrks) Mtrx

14 [99] () Let d B e two squre mtres of the sme order If B B, show tht ( B) B for y postve teger (4 mrks) p q () Let where, re ot oth zero If B, r s show tht B B f d oly f p r d q s (4 mrks) x y () Let C where x, y, z re o-zero d dstt z Fd o-zero mtres D d E suh tht C = D + E d DE = ED = ( mrks) (d) Evlute 5 99 (4 mrks) [] Let M = where Show y duto tht M [ ( ) ] [ ( )] for ll postve tegers Hee or otherwse, evlute (5 mrks) 4 [] () Let e mtrx suh tht I, where I s the detty mtrx () Prove tht hs verse, d fd terms of 4 () Prove tht I () Prove tht I (v) Fd vertle mtrx B suh tht () Let X B B BI (6 mrks) () Usg ()() or otherwse, fd X () Let e postve teger Fd X () Fd two mtres Y d Z, other th X, suh tht Y Y Y I, Z Z Z I (9 mrks) Mtrx 4

15 [] () If det, fd the two vlues of ( mrks) () Let d e vlues oted (), where < Fd d suh tht os os,, s, s Let os os P Evlute P, where s postve teger s s Prove tht P P s mtrx of the form d (8 mrks) d () Evlute, where s postve teger (5 mrks) k k [4] Let =, where,, k R wth k k Defe X ( I) d Y ( I), where I s the detfy mtrx () Evlute XY, YX, X + Y, X d Y (4 mrks) () Prove tht = X + Y for ll postve tegers (4 mrks) () Evlute (4 mrks) (d) If d re o-zero rel umers, guess expresso for - terms of,, X d Y, d verfy t ( mrks) m m [6] Let M where m m m () Evlute M ( mrk) () Let X e o zero rel mtrx suh tht MX XM d () Prove tht d d () Prove tht X s o sgulr mtrx () Suppose tht X 6X () Fd X () If d ( M kx ) M, express k terms of m ( mrks) () Usg the result of ()()(), fd two rel mtres P d Q, other th M d M, suh tht P Q M (4 mrks) Mtrx 5

16 p q [5] () Let M, where p, q, r, s R r s () Suppose det M Prove tht M ( p s) M for y postve teger () Suppose qr Let d e the roots of the qudrt equto x ( p s) x det M Deote the detty mtrx y I () Prove tht d re two dstt rel umers () Prove tht M ( ) M I () Defe M I d B M I Prove tht B B d det det B Fd rel umers d, terms of d, suh tht M B ( mrks) () Evlute 4, where s postve teger Cddtes my use the ft, wthout proof, tht f X d Y re mtres stsfyg XY YX, the ( X Y ) X Y for y postve teger (4 mrks) [7] Let P e o-sgulr rel mtrx d Q, where d re two dstt rel umers Defe M P QP d deote the detty mtrx y I () Fd rel umers d, terms of d, suh tht M () Prove tht det( M I M I) ( ) (6 mrks) [8] Deote the Idetty mtrx y I p q () Let M, where p, q d r re rel umers r p () Evlute M () Let I e rel umer Prove tht f p qr, the for y postve teger, ( s ) ( s ) ( s ) ( s ) (7 mrks) M si M I 6 () Let () Express the form I, where,, d re rel umers 8 () Evlute () Usg the result of ()(), or otherwse, evlute 8 (8 mrks) Mtrx 6

17 [9] Let M os s () Suppose tht M r, where r d Fd r d ( mrks) s os p () rel mtrx of the form s lled dgol mtrx Fd ll postve tegers q suh tht M s dgol mtrx, d evlute M for suh vlues of ( mrks) () Let e rel mtrx suh tht M M Prove tht ( M )( M M M ) M ( mrks) (d) Deote the detty mtrx y I Is I M sgulr mtrx? Expl your swer ( mrks) (e) Evlute M M M M (6 mrks) 4 [] 8 () Let e rel mtrx d 4 () Prove tht P s o-sgulr mtrx () Evlute P P () Let () For y postve teger, fd d d d suh tht 4 B For y postve teger, fd P, where P P d 5 B B B B d (9 mrks) (6 mrks) Mtrx 7