chapter-01 - matrices and determinants

8/2/2019 Chapter-01 - Matrices and Determinants

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Topic: Algebra Sub-topic: Matrices and Determinants

Very Short Answer Questions

1Obj! Spec!Difflevel

1 2 3 7 20 29 KGiven that A = = 2 5 7 and I : l . ' = 2 5 7 without Easy

3 9 13 3 9 13expanding, prove that A = = A' .

Hint/Key:Take R: 7 R, + 2R;" for A.2

If A = [ n f i n d A A ' .Hint/Key:

A A ' = U

KEasy

2 3 ]4 66 9

3 . d . [ 1in the inverse of the matrix 2Hint/Key:1 [ 0 3 ]6 -2 1

UEasy

4 If A [3 - X Y - 3 ] . al . F' d d:::: 0 2 IS a sc ar matnx. m X an y.HintlKey:x =1 andy=3.

KEasy

5 If A = U ~ l B = [ ~ a find A - y, B.HintlKey:[ : ~]

KEasy

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6 KEasy

HintIKey :IABI=-l.7 [ 1 1 ] 4If A = 1 1 ' then find A . KEasy8 I f A = = [ 0 2 ] is skew symmetric, find x and y.y x+4 KEasy

HintlKey : x = = 4 an~ y = = -2.9 U

E a s ya 1 b+cProve that b 1 c + a =O.c 1 a+b

HintIKey: Expect C: -7 C1 + C2 + C3 and then a = O .10 If A ~ [~] and B ~ [1 6 7]. find AB.

HintIKey:A B ~ [ 1 63 18

UE a s y

11 Find a matrix X such that 2A - 3B +X = 0 whereA=[O 3] B=[-2 1]2 5 ' 0 4

UEasy

HintIKey:X = [ - 6 - 3 ]-4 2

12 A is a matrix of order 3 x 4 and B is a matrix of order 4 x 2. KWhat is the order ofrnatrix AS'?HintIKey: AS' does not exist

E a s y

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13[-1IfA= 3

HintIKey :2 ] B _ _[ - 1 2x + 1 ] .1 andB=A'. Fmdx.1 ' 2xx=1

KE a s y

[6 X - X 2 ] UFind the value for x for which the matrix 3 has no E a s yinverse.HintIKey: x =-2.

14

15[ a c d b ] , B = [0 1 01].ind the adjoint of (AB) where A =

HintIKey :

[: ~ ]Adj (AB) = [~c16 IfB + A = [ 1 - 1 3 ] , B _ A = [ 2 3 2 1 ] . Find B.23434

HintIKey:

Short Answer Questions17 If A = [~ ~ ] and IA'I = 125, find the value ofa.

HintIKey:IAI=a2-4Given, 1 A31 = 125 => 1 A 13 = 125(a3 -4/ =125a2-4= 5a=3

18 If A and B are square matrices of the same order andA, then evaluate AB and BA.HintIKey:Given, B" 1 =AOperating B on both sides, we get AB =BA = 1.

KE a s y

KE a s y

KAverage

S-1 = KE a s y

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19 abc a+2b b cIf Al = x Y Z , A2 = x+2y Y z and also Al : ; < ! : 0,

P q r p+2q q rAA2 ' * 0, then find _A2

HintIKey:In A2 , operate el 7 C, - 2C2to get Al = A2

20 If A = [ 2 -1] , prove that A2 - 4A + 31= O.-1 2HintIKey:Find A2, Then substitute in A2 - 4A + 31.RHS=O,

21

HintIKey :2x = -, y= -2,3

22If2A+B~ [~ !~l3A +2B~ [~ ~ afind A' and B',HintIKey:

A ' = [~ ~ ] , B ' = [~ _ 1 6 ]1 - 5 -1 10

23

HintIKey:x = I, Y= 1 and z =-3.

24 For positive values ofx,y, z evaluate1 log",y log", z

logy x 1 logy zlog, X log, y 1

KEasy

KEasy

UEasy

KEasy

KEasy

uDifficult

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HintIKey:Use properties of logarithms.ORReplace1 -7 log, x in R!1 -7 log.y in R21 -7 log.z in R)and use properties of determinant.

25 UAverage

HintlKey:Use mathematical induction

26is independent of 9.

x sine cose UAverageProve that - sin B - x Icose' 1 x

HintIKey:By expanding such that result does not contain 8.

27 [ 1 - 1 ] [ 2fA= andB=2 3 -1HintIKey :(AB)"i = . ! . [ 05 1

KEasy], find (AB)"!.-2

28is singular. Find x.

UEasyIfthe matrix

HintIKey: x =329 Find the eigen values of the matrices

a) [-83

-72]. b) [~ :]

HintIKey:a) 1..=-1,1. .=5b ) 1 ..= -1 ,1 ..= 5

KEasy

30 Solve by using Cramer's rule: x + 3y =5, 2x - y = 3. KEasyHintIKey: x =2 , Y = 1

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31 Solve the equations using matrix method5x + 2y =4, 7x + 3y = 5HintIKey:x =2, y = -3.

KEasy

32 If A is a square matrix of order 2 x2, then show that AA' and UA' A are both symmetric. Easy

If A ~ [~ !]nd B ~ [-24 :3] prove that AB is a null : t . . ymatrix.

33

34

35If MBC of [ l+s~nA

sin A + sin? AProve that AABC is isosceles.HintIKey:

1l+sinB

sinB+sin2 B1 ]+sinC =0

sin Cu- sin ' C

/:1= [ S ~ A S ~ B S i ~ C ]sin 2 A sin 2 B sin 2 C

=(sin A - sin B) (sin B - sin C) (sin C - sin A) =0~A=BorB=C.

KAverage

UAverage

36 If A is amatrix of order 3 x 3 and det IA 1 = 6, find IA-II and KI adj A I . EasyHintIKey: IA-I I = 6 and I adj (A) I =36.

37 Ifthe characteristic equation of a matrix is ' ) , , 1 + 3 A - 2 =0. FindAI in terms of A.

38 1 cosA sinAProve that 1 cos B sin B =

1 cosC sinC. B - C . C - A . A-B4 sm . Sin . sm --2 2 2

UAverageUAverage


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Long Answer questions39 Ifthe maximum and minimum values of determinant A

l+sin" x cos? x sin2x Averagesin? x 1+cos ' x sin2x are a and p ,sin? x cos' x l+sin2x

find a2 + p 2 .HintlKey:C1 7 C1 +C2R2 -7 R2-R.:. d= 2 + sin 2x

40 Verify Cayley Hamilton theorem [: ~ll Hence find itsinverse.HintlKey:Inverse ~ ~ [~l:]

AAverage

41 Solve by matrix method 3x - y + 2z =13, 2x + Y- z =3, x + U3y - 5z =-8. AverageHintIKey:x =3, Y =-2 , z = 1

42 Using Cayley Hamilton theorem, find the inverse ofA =[~ -2

2 _ 1]. Hence find A3 0 -2HintIKey:Characteristic roots ' ) . . _ J - ' ) . . ? - 4 t . . + U = O.:. A3 =A2 + 4A-4 IA-I = -1 [A2-A-4I]

4-148o

UAverage

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43V,erify Cayley Hamilton Theorem for [ ~

-2

13-4

HintIKey:Find characteristic equation.Verify C.H.T. substituting A by A.

44 -be b2 +bc e2 + beProve that a2 +ae -ae e2 +ae = = (ab + bc + ca)3

a2 +bc b2 +ab - abHintlKey: Multiply RJ R2 R3 by a . b. c respectively.Taking a, b, c from Cr, C2, C3.

45[

y+z xFind k if y z +xz z

; ] = k ( X Y Z )x+y

HintIKey: k =-4Evaluate using properties of determinants.

46[ 1 1 1 ]IfA = 1 1 1 , show that A2=3A hence deduce that A4 =1 1 1

27A and A' -A =2A (A + I).HintlKey :Multiplication of A and B =3A.

:. A4=9A2=27AA3 -A=A (A2)-A = 2A2 + 3A-A=2A (A + I)

47 (b+C)2 ab eaProve that ab (e+a)2 be =2 abc (a+b+c)3

ca be (a + b)2HintIKey:Hint: Multiply RJ R2R3by a,b,c and taking abc from cr, C2,C3

48

UAverage

UDifficult

UEasy

UAverage

UAverage

uDifficult

KEasy

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50

51

52

HintIKey:

Find A2 , then A3 = [ 537

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1 a a2 - beProve that 1 b b 2 ca = O.

1 c e2-abHintIKey:R2 ~ R2-RR3 ~ R3-RJ

[cosx

IfF(x) = s~x-smxcosx

o ~how that F(x) F(y) ~ F(x+y)Hint/Key: Show by actual multiplication of two matrices.f(x) . f(y) = Qx + y)

2aProve that

a-b-c2b2c

2ab - c - a 2b = (a+ b + c/

2e 2c

a+b b+e e+a cabProve that c + a a +b b +c = 2 b c a withoutexpansion.

b+c c+a a+b abc

Hint/Key: C1 7 CJ + C2 + C3Take 2 from CJ then C1 7 C1 - C2 ..53 x + 1 x + 2 x + aShow x+2 x+3 x+b =Owherea,b,careinA.P.

x+3 x+4 x+cHintIKey: R2 ~ 2R2 then R2 ~ R2 - (R\ + R3)

54 Ifa,h,c are pth, qth, rth terms ofG.P. Prove that

UAverage

KEasy

UAverage

UAverage

ADifficult

uDifficult

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1 p loga1 q 10gb = 01 r loge

Hint/Key:a =ARP-Jb =ARq-lC = ARr-1Apply log substitute log a log b.

55 UAverage-be ab+ac ac+abProvethat ab+bc -ae bc+ab = (ab +bc s-acj'ac+bc bc+ ac -ab

56 Find the general solution for x.1 cos 2x sin 2x1 cos5x sin5x = O .1 cos7x sin 7x

AAverage

Hi t/K 4' . 5x . 3x 0lD ey : sm x . sm - . sm - =2 2.'. x = nn, 2 n 1{ , 2 n 1{ , n E Z.5 3

57 p b e uIfa#- b, b q. c r, and a q e =0, then find the value of Difficult

a b rp q r--+--+--.p -a q - b r - c

HintIKey:Operate Rll -7 R J -7 R2 and then R~ -7 R2 -7 R3 to getb-a b-q 0

o q-b c-r =0a b r

Expanding along C1 and simplify to getr b a-- +-- +-- =0r - c q - b p-aAdd '2' on both sides, to get the value 2.

chapter-01 - matrices and determinants

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