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1 FREE Dolod Study Pkge from esite:.tekolsses.om &.MthsBySuhg.om Get Solutio of These Pkges & Ler y Video Tutorils o.mthsbysuhg.om SHORT REVISION. Defiitio : Retgulr rry of m umers. Ulike determits it hs o vlue A = : : : : or : : : : m m... m m m... m Arevited s : A = i j i m ; j, i deotes the ro d j deotes the olum is lled mtrix of order m.. Speil Type Of Mtries : () Ro Mtrix :A = [,,... ] hvig oe ro. ( ) mtrix.(or ro vetors) () olum Mtrix : A = : hvig oe olum. (m ) mtrix (or olum vetors) () Zero or Null Mtrix : (A = O m ) m A m mtrix ll hose etries re zero. A = is ull mtrix & B = is ull mtrix (d) Horizotl Mtrix : A mtrix of order m is horizotl mtrix if > m. 5 5 (e) Veril Mtrix : A mtrix of order m is vertil mtrix if m >. (f) Squre Mtrix : (Order ) If umer of ro = umer of olum 6 squre mtrix. Note (i) I squre mtrix the pir of elemets ij & j i re lled ojugte Elemets. (ii) e.g. The elemets,,,... re lled Digol Elemets. The lie log hih the digol elemets lie is lled " Priipl or Ledig " digol. The qty i i = tre of the mtrie ritte s, i.e. t r A Trigulr Mtrix Squre Mtrix Digol Mtrix deote s d di (d, d,..., d ) ll elemets exept the ledig digol re zero digol Mtrix Uit or Idetity Mtrix Note: Mi. umer of zeros i digol mtrix of order = ( ) "It is to e oted tht ith squre mtrix there is orrespodig determit formed y the elemets of A i the sme order.". Equlity Of Mtries : Let A = [ i j ] & B = [ i j ] re equl if, (i) oth hve the sme order. (ii) i j = i j for eh pir of i & j.. Alger Of Mtries : Additio : A + B = i j i j here A & B re of the sme type. (sme order) Teko lsses, Mths : Suhg R. Kriy (S. R. K. Sir), Bhopl Phoe : , pge of 5 Suessful People Reple the ords like; "ish", "try" & "should" ith "I Will". Ieffetive People do't.

2 FREE Dolod Study Pkge from esite:.tekolsses.om &.MthsBySuhg.om Get Solutio of These Pkges & Ler y Video Tutorils o.mthsbysuhg.om () Additio of mtries is ommuttive. i.e. A + B = B + A A = m ; B = m () Mtrix dditio is ssoitive. (A + B) + = A + (B + ) () Additive iverse. Note : A, B & re of the sme type. If A + B = O = B + A A = m 5. Multiplitio Of A Mtrix By A Slr : If A = k k k ; k A = k k k k k k 6. Multiplitio Of Mtries : (Ro y olum) AB exists if, A = m & B = p AB exists, ut BA does ot AB BA Note : I the produt AB, A preftor B post ftor A = (,,... ) & B = A B = [ ] If A = i j m & B = i j p mtrix, the (A B) = i j r Properties Of Mtrix Multiplitio :. Mtrix multiplitio is ot ommuttive. A = : ; B = ;AB = ; BA = AB BA (i geerl) i r. r j. AB = = AB = O A = O or B = O Note: If A d B re to o- zero mtries suh tht AB = O the A d B re lled the divisors of zero. Also if [AB] = O AB A B = A = or B = ut ot the overse. If A d B re to mtries suh tht (i) AB = BA A d B ommute eh other (ii) AB = BA A d B ti ommute eh other. Mtrix Multiplitio Is Assoitive : If A, B & re oformle for the produt AB & B, the (A. B). = A. (B. ). Distriutivity : A ( B ) AB A ( A B) A B Provided A, B & re oformle for respetive produts 5. POSITIVE INTEGRAL POWERS OF A SQUARE MATRIX : For squre mtrix A, A A = (A A) A = A (A A) = A. Note tht for uit mtrix I of y order, I m = I for ll m N. Suessful People Reple the ords like; "ish", "try" & "should" ith "I Will". Ieffetive People do't. Teko lsses, Mths : Suhg R. Kriy (S. R. K. Sir), Bhopl Phoe : , pge of 5

3 FREE Dolod Study Pkge from esite:.tekolsses.om &.MthsBySuhg.om Get Solutio of These Pkges & Ler y Video Tutorils o.mthsbysuhg.om 6. MATRIX POLYNOMIAL : If f (x) = x + x + x x the e defie mtrix polyomil f (A) = A + A + A I here A is the give squre mtrix. If f (A) is the ull mtrix the A is lled the zero or root of the polyomil f (x). DEFINITIONS : () Idempotet Mtrix : A squre mtrix is idempotet provided A = A. Note tht A = A >, N. () Nilpotet Mtrix: A squre mtrix is sid to e ilpotet mtrix of order m, m N, if A m = O, A m O. () Periodi Mtrix : A squre mtrix is hih stisfies the reltio A K+ = A, for some positive iteger K, is periodi mtrix. The period of the mtrix is the lest vlue of K for hih this holds true. Note tht period of idempotet mtrix is. (d) Ivolutry Mtrix : If A = I, the mtrix is sid to e ivolutry mtrix. Note tht A = A for ivolutry mtrix. 7. The Trspose Of A Mtrix : (hgig ros & olums) Let A e y mtrix. The, A = of order m i j A T or A = [ j i ] for i & j m of order m Properties of Trspose : If A T & B T deote the trspose of A d B, () (A ± B) T = A T ± B T ; ote tht A & B hve the sme order. IMP. () (A B) T = B T A T A & B re oformle for mtrix produt AB. () (A T ) T = A (d) (k A)T = k AT k is slr. Geerl : (A, A,... A ) T = A T,..., A T, A T (reversl l for trspose) 8. Symmetri & Ske Symmetri Mtrix : A squre mtrix A = i j is sid to e, symmetri if, i j = j i i & j (ojugte elemets re equl) (Note A = A T ) ( ) Note: Mx. umer of distit etries i symmetri mtrix of order is. d ske symmetri if, i j = j i i & j (the pir of ojugte elemets re dditive iverse of eh other) (Note A = A T ) Hee If A is ske symmetri, the i i = i i i i = i Thus the digol elemets of ske symmetri mtrix re ll zero, ut ot the overse. Properties Of Symmetri & Ske Mtrix : P A is symmetri if A T = A A is ske symmetri if A T = A P A + A T is symmetri mtrix A A T is ske symmetri mtrix. osider (A + A T ) T = A T + (A T ) T = A T + A = A + A T A + A T is symmetri. Similrly e prove tht A A T is ske symmetri. P The sum of to symmetri mtrix is symmetri mtrix d the sum of to ske symmetri mtrix is ske symmetri mtrix. Let A T = A ; B T = Bhere A & B hve the sme order. (A + B) T = A + B Similrly e prove the other Teko lsses, Mths : Suhg R. Kriy (S. R. K. Sir), Bhopl Phoe : , pge of 5 Suessful People Reple the ords like; "ish", "try" & "should" ith "I Will". Ieffetive People do't.

4 FREE Dolod Study Pkge from esite:.tekolsses.om &.MthsBySuhg.om Get Solutio of These Pkges & Ler y Video Tutorils o.mthsbysuhg.om P If A & B re symmetri mtries the, () A B + B A is symmetri mtrix () AB BA is ske symmetri mtrix. P 5 Every squre mtrix e uiquely expressed s sum of symmetri d ske symmetri mtrix. 9. Adjoit Of A Squre Mtrix : Let A = i j = A = (A + AT ) + (A AT ) P Q Symmetri Ske Symmetri oftors of [ i j ] i determit A is = The (dj A) = e squre mtrix d let the mtrix formed y the V. Imp. Theorem : A (dj. A) = (dj. A).A = A I, If A e squre mtrix of order. Note : If A d B re o sigulr squre mtries of sme order, the (i) dj A = A (ii) dj (AB) = (dj B) (dj A) (iii) dj(ka) = K (dj A), K is slr Iverse Of A Mtrix (Reiprol Mtrix) : A squre mtrix A sid to e ivertile (o sigulr) if there exists mtrix B suh tht, A B = I = B A B is lled the iverse (reiprol) of A d is deoted y A. Thus We hve, A = B A B = I = B A. A. (dj A) = A I A A (dj A) = A I I (dj A) = A A I A = ( dj A ) A Note : The eessry d suffiiet oditio for squre mtrix A to e ivertile is tht A. Imp. Theorem : If A & B re ivertile mtries ofthe sme order, the (AB) = B A. This is reversl l for iverse. Note :(i) If A e ivertile mtrix, the A T is lso ivertile & (A T ) = (A ) T. (ii) If A is ivertile, () (A ) = A ; () (A k ) = (A ) k = A k, k N (iii) If A is Orthogol Mtrix. AA T = I = A T A (iv) A squre mtrix is sid to e orthogol if, A = A T. (v) A = SYSTEM OF EQUATION & RITERIAN FOR ONSISTENY GAUSS - JORDAN METHOD x + y + z = 6 x y + z = Suessful People Reple the ords like; "ish", "try" & "should" ith "I Will". Ieffetive People do't.. A Teko lsses, Mths : Suhg R. Kriy (S. R. K. Sir), Bhopl Phoe : , pge of 5

5 Get Solutio of These Pkges & Ler y Video Tutorils o.mthsbysuhg.om x + y z = FREE Dolod Study Pkge from esite:.tekolsses.om &.MthsBySuhg.om or xyz xyz xyz 6 = x 6 y = z A X = B A A X = A B X = A B = (dj. A).B. A Note :() If A, system is osistet hvig uique solutio () If A & (dj A). B O (Null mtrix), system is osistet hvig uique o trivil solutio. () If A & (dj A). B = O (Null mtrix),system is osistet hvig trivil solutio. () If A =, mtrix method fils Q. Give tht A = If (dj A). B = ull mtrix = O osistet (Ifiite solutios), = If (dj A). B O Iosistet (o solutio) EXERISE- Ax =. Q. Fid the vlue of x d y tht stisfy the equtios. Q. If, E = Q. y x y x = y d F = y, D = d tht = D. Solve the mtrix equtio 9 lulte the mtrix produt EF & FE d sho tht E F + FE = E. If A is orthogol mtrix d B = AP here P is o sigulr mtrix the sho tht the mtrix PB is lso orthogol. Q 5. The mtrix, R(t) is defied y R(t) = ost sit sit. Sho tht, R(s) R(t) R(s + t). ost os si os os Q 6. Prove tht the produt of to mtries, ossi si & ossi mtrix he & differ y odd multiple of. sios is ull si Teko lsses, Mths : Suhg R. Kriy (S. R. K. Sir), Bhopl Phoe : , pge 5 of 5 Suessful People Reple the ords like; "ish", "try" & "should" ith "I Will". Ieffetive People do't.

6 FREE Dolod Study Pkge from esite:.tekolsses.om &.MthsBySuhg.om Get Solutio of These Pkges & Ler y Video Tutorils o.mthsbysuhg.om Q 7. If, A =, the sho tht the mxtrix A is root of the polyomil f (x) = x 6x + 7x +. Q.8 For o zero, use idutio to prove tht : (Oly for XII BSE) () = ( ), for every N () If, A =, the (I + A) = I + A, here I is uit mtrix of order, N. Q9. Fid the umer of mtrix stisfyig (i) ij is or ; (ii) + = + = ; (iii) + = Q. Prove tht (AB) T = B T. A T, here A & B re oformle for the produt AB. Also verify the result Q for the mtries, A = Express the mtrix d B = Suessful People Reple the ords like; "ish", "try" & "should" ith "I Will". Ieffetive People do't s sum of loer trigulr mtrix & upper trigulr mtrix ith zero i its ledig digol. Also Express the mtrix s sum of symmetri & ske symmetri mtrix. Q. Fid the iverse of the mtrix : os (i) A = si (iii) A = si os ; (ii) here is the ue root of uity. Q. Fid the mtrix A stisfyig the mtrix equtio,. A. 5 =. Q. A is squre mtrix of order. l = mximum umer of distit etries if A is trigulr mtrix m = mximum umer of distit etries if A is digol mtrix p = miimum umer of zeroes if A is trigulr mtrix If l + 5 = p + m, fid the order of the mtrix. Q 5. If A is idempotet mtrix d I is idetity mtrix of the sme order, fid the vlue of, N, suh tht ( A + I ) = I + 7 A. Q.6 If A = d the prove tht vlue of f d g stisfyig the mxtrix equtio A + f A + g I = O re equl to t r (A) d determit of A respetively. Give,,, d re o zero rels d I = ; O =. Teko lsses, Mths : Suhg R. Kriy (S. R. K. Sir), Bhopl Phoe : , pge 6 of 5

7 FREE Dolod Study Pkge from esite:.tekolsses.om &.MthsBySuhg.om Get Solutio of These Pkges & Ler y Video Tutorils o.mthsbysuhg.om Q7. Mtries A d B stisfy AB = B here B =. Fid (i) ithout fidig B, the vlue of K for hih KA B + I = O (ii) Without fidig A, the mtrix X stisfyig A XA = B (iii) the mtrix A, usig A Q8. For the mtrix A = Q9. Give A = 5 fid A., B =. Fid P suh tht BPA = Q. Use mtrix to solve the folloig system of equtios. (i) xyz xyz xy9z6 (ii) xyz6 xyz xyz (iii) xyz xyz xyz7 (iv) EXERISE-5 xyz xyz xyz9 Q. 9 Give A = ; B =. I is uit mtrix of order. Fid ll possile mtrix X i the folloig ses. (i) AX = A (ii) XA = I (iii) XB = O ut BX O. Q. If A & B re squre mtries of the sme order & A is symmetril, sho tht B AB is lso symmetril. t Q. Sho tht, t t t Q. If the mtries A = d B = d Q 5. os si = si os. (,,, d ot ll simulteously zero) ommute, fid the vlue of mtrix hih ommutes ith A is of the form d. Also sho tht the If the mtrix A is ivolutry, sho tht (I + A) d (I A) re idempotet d (I + A) (I A)=O. Q 6. Prove tht (i) dj (dj A) = (ii) ( A ), here A is o-sigulr mtrix of order ''. dj (dj A) = A. A, here A deotes the determit of o-effiiet mtrix. 5 Q 7. Fid the produt of to mtries A & B, here A = 7 solve the folloig system of lier equtios, x + y + z = ; x + y + z = 7 ; x + y + z =. 5 & B = d use it to Teko lsses, Mths : Suhg R. Kriy (S. R. K. Sir), Bhopl Phoe : , pge 7 of 5 Suessful People Reple the ords like; "ish", "try" & "should" ith "I Will". Ieffetive People do't.

8 FREE Dolod Study Pkge from esite:.tekolsses.om &.MthsBySuhg.om Get Solutio of These Pkges & Ler y Video Tutorils o.mthsbysuhg.om Q 8. If A = the, fid o-zero squre mtrix X of order suh tht AX = O. Is XA = O. If A =, is it possile to fid squre mtrix X suh tht AX = O. Give resos for it. os si os si Q 9. If A = si os ; B= si os Where the prove tht BAB = A. Also fid the lest positive vlue of for hih B A B = A. Q. If is idempotet mtrix. Fid the vlue of f(), here f(x) = x x, he = /. Hee otherise evlute. Q. If A is ske symmetri mtrix d I + A is o sigulr, the prove tht the mtrix B = (I A)(I + A) 5 is orthogol mtrix. Use this to fid mtrix B give A = 5. os x Q. If F(x) = si x si x os x Hee prove tht [ F(x) ] = F( x). the sho tht F(x). F(y) = F(x + y) Q. If A = ; B = ; = x x d X = x x the solve the folloig mtrix equtio. () AX = B I () (B I)X = I () X = A Q. Determie the vlues of d for hih the system 5 x 9 y z Suessful People Reple the ords like; "ish", "try" & "should" ith "I Will". Ieffetive People do't. 8 (i) hs uique solutio ; (ii) hs o solutio d (iii) hs ifiitely my solutios Q 5. Let X e the solutio set of the equtio A x = I, here A = uit mtrix d x N the fid the miimum vlue of (os Q6. Determie the mtries B d ith itegrl elemet suh tht A = Q7. If A = Q8.If A = k Q9. Evlute l = B + m d Lim x x d I is the orrespodig x si ), R. is orthogol mtrix, fid the vlues of,,. k lm ; the sho tht A (k + )A + (k lm) I = O.Hee fid A. x Teko lsses, Mths : Suhg R. Kriy (S. R. K. Sir), Bhopl Phoe : , pge 8 of 5

9 FREE Dolod Study Pkge from esite:.tekolsses.om &.MthsBySuhg.om Get Solutio of These Pkges & Ler y Video Tutorils o.mthsbysuhg.om Q. Give mtries A = x x y y ; B = z z Oti x, y d z if the mtrix AB is symmetri. EXERISE-6 Q. If mtrix A = here,, re rel positive umers, = d A T A = I, the fid the vlue of + +. [JEE, Mis- out of 6] Q. If A = d the = (A) (B) () 5 (D) [JEE (Sreeig)] Q. If M is mtrix, here M T M = I d det (M) =, the prove tht det (M I) =. Q. A = d, B = f d g, U = h f g, V = h. If there is vetor mtrix X, suh tht AX = U hs ifiitely my solutio, the prove tht BX = V ot hve uique solutio. If fd, the prove tht BX = V hs o solutio. Q.5 A =, I = d A = (A 6 A di), the the vlue of d d re (A) 6, (B) 6, () 6, (D) 6, Q.6 If P = (A) 5, A = d Q = PAP T d x = P T Q 5 P, the x is equl to 5 (B) () (D) 5 Q7. If f(x) is qudrti polyomil d,, re three rel d distit umers stisfyig f ( ) f () f (). Give f(x) uts the x-xis t A d V is the poit of mixim. If AB is y hord hih suteds right gle t V, fid urve f(x) d re ouded y hord AB d urve f(x). Teko lsses, Mths : Suhg R. Kriy (S. R. K. Sir), Bhopl Phoe : , pge 9 of 5 Suessful People Reple the ords like; "ish", "try" & "should" ith "I Will". Ieffetive People do't.

10 Suessful People Reple the ords like; "ish", "try" & "should" ith "I Will". Ieffetive People do't. Teko lsses, Mths : Suhg R. Kriy (S. R. K. Sir), Bhopl Phoe : , pge of 5 FREE Dolod Study Pkge from esite:.tekolsses.om &.MthsBySuhg.om Get Solutio of These Pkges & Ler y Video Tutorils o.mthsbysuhg.om A, if U, U d U re olums mtries stisfyig. AU =, AU =, AU = d U is x mtrix hose olums re U, U, U the ser the folloig questios. Q8. The vlue of U is [JEE 6] (A) (B) - () / (D) Q9. The sum of the elemets of U - is [JEE 6] (A) - (B) () (D) Q. The vlue of U is [JEE 6] (A) 5 (B) 5/ () (D) / ANSWER SHEET EXERISE- Q. x =, x =, x = Q. x =, y = Q. EF =, FE = Q.9 8 Q ; + Q. (i) os si si os, (ii), (iii) Q Q. Q.5 = 7 Q.6 f = ( + d) ; g = d Q.7 (i) K =, (ii) X = B, (iii) A = Q Q Q. (i) x =, y =, z = ; (ii) x =, y =, z = ; (iii) x = + k, y = k, z = k here k R ; (iv) iosistet, hee o solutio

11 FREE Dolod Study Pkge from esite:.tekolsses.om &.MthsBySuhg.om Get Solutio of These Pkges & Ler y Video Tutorils o.mthsbysuhg.om EXERISE-5 Q. (i) X = for, R ; (ii) X does ot exist. ; (iii) X =, R d + ; + d d Q. Q.7 x =, y =, z = Q.8 X = d, here, d R {}, NO Q.9 Q.() X= 5 Q. f () = /, = / Q., () X =, () o solutio Q. (i), R ; (ii) = d / ; (iii) =, = / 5 Q.5 Q.6 B = d = Q.7,, 6 os x si x Q.9 si x os x m Q.8 k lm l k Q.,,,,,, (,, ) EXERISE-6 Q. Q. A Q.5 Q.6 A Q7. Q9. B Q. A 5 5 sq. uits Q.8 A Teko lsses, Mths : Suhg R. Kriy (S. R. K. Sir), Bhopl Phoe : , pge of 5 Suessful People Reple the ords like; "ish", "try" & "should" ith "I Will". Ieffetive People do't.