Free Matrices 02 Practice Test - 12th Grade - Commerce
Question 1
If A=[abba] and A2=[αββα], then
A.
α=a2+b2,β=ab
B.
α=a2+b2,β=2ab
C.
α=a2+b2,β=a2−b2
D.
α=2ab,β=a2+b2
SOLUTION
Solution : B
A2=[αββα]=[abba][abba];α=α2+b2;β=2ab
Question 2
If A=⎡⎣1tanθ2−tanθ21⎤⎦ and AB = I, then B =
A.
cos2θ2.A
B.
cos2θ2.AT
C.
cos2θ2.I
D.
None of these
SOLUTION
Solution : B
|A|=1+tan2θ2=sec2θ2AB=I⇒B−IA−1[1001]⎡⎢⎣1−tanθ2tanθ21⎤⎥⎦sec2θ2=cos2θ2.AT.
Question 3
Let A=⎡⎢⎣46−13021−25⎤⎥⎦,B=⎡⎢⎣2401−12⎤⎥⎦and C = [3 1 2]. The expression which is not defined is
A.
B'B
B.
CAB
C.
A+B'
D.
A2+A
SOLUTION
Solution : C
We can see from the options that if we take transpose of B, B' will be of 2 x 3 matrix which cannot be added to a 3 x 3 matrix, as for the addition the order should be same.
Question 4
If A is 3×4 matrix and B is a matrix such that A'B and BA' are both defined. Then B is of the type
A.
3×4
B.
3×3
C.
4×4
D.
4×3
SOLUTION
Solution : A
A3×4⇒A′4×3Now A'B defined
⇒Bis 3×p
Again B3×pA′4×3 defined ⇒p=4
∴B is 3×4.
Question 5
If A=⎡⎢⎣a000b000c⎤⎥⎦, then An=
A.
⎡⎢⎣na000nb000nc⎤⎥⎦
B.
⎡⎢⎣a000b000c⎤⎥⎦
C.
⎡⎢⎣an000bn000cn⎤⎥⎦
D.
None of these
SOLUTION
Solution : C
Since A2=A.A=⎡⎢⎣a000b000c⎤⎥⎦⎡⎢⎣a000b000c⎤⎥⎦=⎡⎢⎣a2000b2000c2⎤⎥⎦
And A3=⎡⎢⎣a3000b3000c3⎤⎥⎦,....⇒An=An−1.A=⎡⎢⎣an−1000bn−1000cn−1⎤⎥⎦⎡⎢⎣a000b000c⎤⎥⎦=⎡⎢⎣an000bn000cn⎤⎥⎦.
Note: Students should remember this question as a formula.
Question 6
For each real number x such that −1<x<1,let A(x) be the matrix (1−x)−1[1−x−x1] and z=x+y1+xyThen,
A.
A(z)=A(x)+A(y)
B.
A(z)=A(x)+[A(y)]−1
C.
A(z)=A(x)A(y)
D.
A(z)=A(x)–A(y)
SOLUTION
Solution : C
A(z)=A(x+y1+xy)=[1+xy(1−x)(1−y)] ⎡⎢⎣1−(x+y1+xy)−(x+y1+xy)1⎤⎥⎦
∴A(x).A(y)=A(z)
Question 7
If A=[cosθsinθsinθ−cosθ],B=[10−11],C=ABAT,then ATCnA equals to(nϵZ+)
A.
[−n110]
B.
[1−n01]
C.
[011−n]
D.
[10−n1]
SOLUTION
Solution : D
A=[cosθsinθsinθ−cosθ]AAT=I (i)Now,C=ABAT⇒ATC=BAT (ii)Now ATCnA=ATC.Cn−1A=BATCn−1A(from(ii))=BATC.Cn−2A=B2ATCn−2A=.......=Bn−1ATCA=Bn−1BATA=Bn=[10−n1]
Question 8
A=[aij]n×n and aij=i2−j2 then A is necessarily
A.
a unit matrix
B.
symmetric matrix
C.
skew symmetric matrix
D.
zero matrix
SOLUTION
Solution : C
aji=j2−i2=−(i2−j2)=−aij
Question 9
If A is a non-diagonal involutory matrix, then
A.
A - I = 0
B.
A + I = 0
C.
A - I is non zero singular
D.
none of these
SOLUTION
Solution : C
A2=I⇒A2−I=0
⇒ (A+I)(A-I)=0
∴ either |A+I|=0 or
|A−I|=0
If |A−I|≠0, then (A+I)(A−I)=0⇒A+I=0 which is not so
∴|A−I| and A−I≠0.
Question 10
If A and B are two non singular matrices and both are symmetric and commute each other then
A.
Both A−1B and A−1B−1 are symmetric
B.
A−1B is symmetric but A−1B−1 is not symmetric
C.
A−1B−1 is symmetric but A−1B is not symmetric
D.
Neither A−1B nor A−1B−1 are symmetric
SOLUTION
Solution : A
AB =BA
Previous & past multiplying both sides by A−1.
A−1(AB)A−1=A−1(BA)A−1(A−1A)(BA−1)=A−1B(AA−1)⇒(BA−1)1=(A−1B)1=(A−1)1B1(reversal laws)=A−1B(as B=B1)(A−1)1=A−1⇒A−1B is symmetric
Similarly for A−1B−1.