UNIT 2: MATRICES MATRICES AND DETERMINANTS
Choose the correct answer a)Every scalar matrix is an identity matrix b)Every b)Ever y identity matrix is a scalar matrix matrix c)Every diagonal matrix is an identity matrix d)A square matrix whose each element is an identity matrix A=(aij) we find that a ij =a ji for 2. If in a square matrix A=(a all i j then A is a)symmetric matrix b)traiangular b)traiangular matrix c)trans!ose c)trans!o se matrix d)s"ew symmetric 1.
3.
4.
5.
6.
8.
c) 14.
#hich of the following is not correct for a traingular matrix A=(aij)$
c) 15.
%
f A is a s!"a#e $at#ix then AA is a)s&$$et#ic b)sew s&$$et#ic c)scala# c)scala# $at#ix d)"nit $at#ix f a $at#ix A is s&$$et#ic and sew s&$$et#ic then A is a a)diagonal $at#ix b)n"ll $at#ix
16.
d),one of the above
/
1
1
−1
4
0
2
5
2
/
%
5
%
4
10.
is
%
−1 −0
c)
f the $at#ix
a b d
a)
0
1
5
1
/
%
5
%
4
−1 −0 % −1 −1 %
1
−5
c e
-
f the $at#ix
[
x 2
-
]
is s&$$et#ic, s &$$et#ic,
x is a)- b)*- c). d) 2 12. Ass"$ing Ass"$ing that the s"$ and #od"cts gi0en below a#e defined which of the following is not t#"e fo# $at#ices? a)AB=AC does not i$l& B=C(A*- exist) b)(AB)' =B'A' c)A+B=B+A d)AB=O A=.o# B=. 1hat $"st be the $at#ix if 2+
3
2 /
=
3 2
?
2 3
.
3
-
4
2
∣
−-
−-
2
−4
/
−2
/
2
and A+2B=
then B=
)
-
2
-
-.
-
-
2
−-
-.
−-
−−2
2 3
−- -.
b)
)
2
d
−2 −-
)
−-
2
−-
-.
−-
] ]
then A/ +A3 *A2 =
%
%
−0
%
−0
%
−0
%
%
ab
-
ab ac
b
2
he only correct
∣
ac bc
-
bc
∣
-.-./ -.
18.
19.
c
2
= a)-+a+b+c
-
∣
-.3 -. -. -.4 -. -.6
∣ ∣ x p p
7=
p x q
q q x
= a). b)- c)2 d)3
= (2..)
a)(x+)(x+!)(x**!) b)(x*)(x*!)(x++!) c)(x*)(x*!)(x**!) d)(x+)(x+!)(x++!) 20.
∣
f a-, a2 , a3 888888888an 8888888 a#e in 9:, then the 0al"e of the dete#$inant
loga n loga n loga n 3 loga n
loga n 21.
4
loga n
/
loga n loga n
loga n
-
∣
2 (
= a)2 b)- c). d)*2
)
%he 0al"e of the dete#$inant
∣
22. 13.
−2
d)
−3
b)0.ab.bc.ca c)0.a 1 .b1 .c1 d)abc
f
/x (
/
6et A=
17.
is a sew
2
4
d),one
s&$$et#ic $at#ix, then a+b+c+d+e+f= a+b+c+d+e+f= a)- b). c)*/ d)-. 11.
2
(
a
[ ] 3
A=
b)
2
−-
f A=
%he s&$$et#ic a#t of the $at#ix 1
2
2
b)
statement about the matrix A is a)A*- does not exist b)A=(*-) c)A is a 5e#o $at#ix d)A2=
If A is a s"ew symmetric matrix then (A- .A/) is equal to a)A +A b)A *A c)A *A c)A% +B% f A is a s&$$et#ic $at#ix and B is a sew s &$$et#ic $at#ix of the sa$e sa$e o#de# , then A2 +B2 is a
0
[ ] [ ] [ ] [ [ ] [ [ ] 3
a). b) c)A d)one of these
a)s&$$et#ic $at#ix b)sew s&$$et#ic $at#ix c)"nit $at#ix d)one 9.
-
f 2A+3B=
a)
a) aij =% for all i&j b)It is not necessary that aij '% for all i≤ j c)A must be square matrix d)he elements in the !rinci!al diagonal may or may not be ero* If O(A)=2x3, O(B)=3x2 and O(C)=3x3, which of the following is not defined? a)C(A+B') b)C(A+B')' c)BAC d)CB+A'
c)+nit matrix 7.
a)
∣
-
-
-
mC -
m-C-
m 2C-
mC 2
m-C2
m2C 2
is e!"al to
a)- b)*- c). d)one of these %he 0al"e of the dete#$inant ;=
∣
∣
-!
2!
3!
2!
3!
/!
3!
/!
(!
is a)2< b)3< c)/< d)<
∣
23.
24.
−-
/
-
.
-
.
.
(
2
3
.
.
−2
2
−3
sing"la#? a)2 b). 25. If
x 2
x 3
−-
2
31.
∣
-
-
-
- c
a
-
b
:=
32.
=., then the 0al"e
-
33.
is
c
d)*a*b*c abc
∣
∣
∣
x −a . x c
x −b x −c .
34.
35.
then f(2.-2) is a). b)- c)2.. d)*2.. (C.) If lmn are !th qth rth term of 7*8 all !ositive
then a)90
∣ [
b)1
If A=
p q r
∣ ]
-
36.
3
-
2
-
−3
-
-
-
cc2 c3
c2 c3 a2 a3 b2 b 3
and
then
[
2
-
/
/
2
−3
-
-
2
]
the 0al"es of $ino#s of the
[
] [ ] [ ] [ ] [ ] . . 3
−2
2 2 −3 −2 /
. .
. .
. .
. . .
. . .
. . .
b)
d)
then A*adj(A) is equal to
(
-
-
-
(
-
-
-
(
. 4 .
. . 4
4 . .
(1%%4)
If A(adjA)=-I where I is identity matrix of order 5 then ;adjA; is equal to a)- b)0% c)01- d)1-(1%%/) If A is a square matrix of order 5 and ;A;=/< then ; )
37.
f A is a 3x3 $at#ix and adoint of A is B8 f B=4/, then A= a)4/ b)/ c) d)3
38.
f x+3&+5=4, 3x*2&*5= and /x+&*35=- , B& $at#ix $ethod x,&,5 a#e a)-,,. b),.,- c),-,. d)one %he e!"ation x+2&+35=-D x*&+/5=.D 2x+&+5=ha0e (CE%6)a)onl& one sol"tion b)onl& two sol"tion c)no sol"tion d)infinitel& $an& sol"tion
39.
c)0 d)% (AIEEE%1) −-
bb2 b3
adjA;= a)/ b)/1 c)/5 d)
equals
-
-
cab-
If A=
c)
∣
logl logm logn
if
=. is
x x x x −- x - x x x −- x − 2 x - x x −-
2x
and : is the inverse of A
a):=% b):=A1 c)A= 9: d)A=:(1%%/) f A and B a#e s!"a#e $at#ices of o#de# 3 s"ch thatA=*-8 B =3 , then 3AB is e!"al to a)*6 b)*- c)*2 d)-
a)
a)x=. b)x=c c)x=b d)x=a If f(x)= -
]
ele$ents of fi#st #ow a#e #esecti0el& a),--,2 b)-,,4 c)2,--, d)--,,2
x y y z z x %he 0al"e of = x y z x − y y − z z − x a). b)(x+&+5) 3 c)2(x+&+5)3 d)2(x+&+5)2 f a@b@c, a #oot of the e!"ation
.
2
∣ ∣ ∣ ∣ aa2 a3
If A=
is
d)-
- b
3x x − -
30.
−-
-
x a x b
29.
3
-
∣
∣
2
c)3
a)*- b)abc c)
28.
]
− - x
-
-
27.
[
3
- a
of
2 . −2
then the value of is a)% b)1 c)2 d)-
a, b and c a#e all diffe#ent f#o$ 5e#o and
∣
[
/ − -
0%:=
=a)/- b)- c)3- d)24
o# how $an& 0al"es of x in the closed inte#0al >*/,*-, the $at#ix
26.
∣
-
(CE1%%4) 40.
f A2*/A* =., then A*- =a)/A*b)A*/ c)A*/F
