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12th Grade > Mathematics

MATRICES MCQs

Total Questions : 44 | Page 2 of 5 pages
Question 11. If A=[cosθsinθsinθ−cosθ],B=[10−11],C=ABAT,then ATCnA equals to(nϵZ+)
  1.    [−n110]
  2.    [1−n01]
  3.    [011−n]
  4.    [10−n1]
 Discuss Question
Answer: Option D. -> [10−n1]
:
D
A=[cosθsinθsinθ−cosθ]AAT=I(i)Now,C=ABAT⇒ATC=BAT(ii)NowATCnA=ATC.Cn−1A=BATCn−1A(from(ii))=BATC.Cn−2A=B2ATCn−2A=.......=Bn−1ATCA=Bn−1BATA=Bn=[10−n1]
Question 12. If A=[abba] and A2=[αββα], then
  1.    α=a2+b2,β=ab
  2.    α=a2+b2,β=2ab
  3.    α=a2+b2,β=a2−b2
  4.    α=2ab,β=a2+b2
 Discuss Question
Answer: Option B. -> α=a2+b2,β=2ab
:
B
A2=[αββα]=[abba][abba];α=α2+b2;β=2ab
Question 13. 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
  1.    3×4
  2.    3×3
  3.    4×4
  4.    4×3
 Discuss Question
Answer: Option A. -> 3×4
:
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 14. 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,
  1.    A(z)=A(x)+A(y)
  2.    A(z)=A(x)+[A(y)]−1
  3.    A(z)=A(x)A(y)
  4.    A(z)=A(x)–A(y)
 Discuss Question
Answer: Option C. -> A(z)=A(x)A(y)
:
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 15. Let A=⎡⎢⎣46−13021−25⎤⎥⎦,B=⎡⎢⎣2401−12⎤⎥⎦and C = [3 1 2]. The expression which is not defined is
  1.    B'B
  2.    CAB
  3.    A+B'
  4.    A2+A
 Discuss Question
Answer: Option C. -> A+B'
:
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 16. If A=⎡⎣1tanθ2−tanθ21⎤⎦ and AB = I, then B =
  1.    cos2θ2.A
  2.    cos2θ2.AT
  3.    cos2θ2.I
  4.    None of these
 Discuss Question
Answer: Option B. -> cos2θ2.AT
:
B
|A|=1+tan2θ2=sec2θ2AB=I⇒B−IA−1[1001]⎡⎢⎣1−tanθ2tanθ21⎤⎥⎦sec2θ2=cos2θ2.AT.
Question 17. If matrix A=[aij]3×3,matrix B=[bij]3×3 where aij+aji=0 and bij−bji=0,then A4.B3 is
  1.    skew-symmetric matrix
  2.    singular
  3.    symmetric
  4.    zero matrix
 Discuss Question
Answer: Option B. -> singular
:
B
Since matrix A is skew-symmetric,
∴|A|=0
∴|A4.B3|=0
Question 18. If A and B are symmetric matrices of same order and X= AB + BA and Y = AB – BA, then (XY)T is equal to
  1.    XY
  2.    YX
  3.    – YX
  4.    None of these
 Discuss Question
Answer: Option C. -> – YX
:
C
X=AB+BA⇒XT=X
andY=AB−BA⇒YT=−Y
Now,(XY)T=YT×XT=−YX
Question 19. If A', B' are transpose matrices of the square matrices A, B respectively , then (AB)' is equal to
  1.    A'B'
  2.    B'A'
  3.    AB'
  4.    BA'
 Discuss Question
Answer: Option B. -> B'A'
:
B
It is a fundamental concept ,i.e., (AB)' = B'A'
Question 20. If A and B are two matrices such that AB = B and BA = A, then
  1.    (A6−B5)3=A−B
  2.    (A5−B5)3=A3−B3
  3.    A−B is idempotent
  4.    A−B is nilpotent
 Discuss Question
Answer: Option D. -> A−B is nilpotent
:
D
Since AB = B and BA = A
∴ A and B both are idempotent
(A−B)2=A2−AB−BA+B2=A−B−A+B=0
∴ A - B is nilpotent

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