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12th Grade > Mathematics

MATRICES MCQs

Total Questions : 44 | Page 1 of 5 pages
Question 1. If A=a000b000c, then An=
  1.    ⎡⎢⎣na000nb000nc⎤⎥⎦
  2.    ⎡⎢⎣a000b000c⎤⎥⎦
  3.    ⎡⎢⎣an000bn000cn⎤⎥⎦
  4.    None of these
 Discuss Question
Answer: Option C. -> ⎡⎢⎣an000bn000cn⎤⎥⎦
:
C
Since A2=A.A=a000b000ca000b000c=a2000b2000c2
And A3=a3000b3000c3,....An=An1.A=an1000bn1000cn1a000b000c=an000bn000cn.
Note: Students should remember this question as a formula.
Question 2. If A and B are two non singular matrices and both are symmetric and commute each other then
  1.    Both A−1B and A−1B−1 are symmetric
  2.    A−1B is symmetric but A−1B−1 is not symmetric
  3.    A−1B−1  is symmetric but A−1B is not symmetric
  4.    Neither A−1B nor A−1B−1 are symmetric
 Discuss Question
Answer: Option A. -> Both A−1B and A−1B−1 are symmetric
:
A
AB =BA
Previous & past multiplying both sides by A1.
A1(AB)A1=A1(BA)A1(A1A)(BA1)=A1B(AA1)(BA1)1=(A1B)1=(A1)1B1(reversallaws)=A1B(asB=B1)(A1)1=A1A1B is symmetric
Similarly for A1B1.
Question 3. 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 4. If A is a skew-symmetric matrix of order 3, then the matrix A4 is
  1.    skew symmetric
  2.    symmetric
  3.    diagonal
  4.    none of those
 Discuss Question
Answer: Option B. -> symmetric
:
B
We are given that the matrix A is skew symmetric which implies that,
AT=A --------(1)
(A4)T=(A.A.A.A.)T=ATATATAT (By applying the property of transpose of a matrix)
(-A) (-A) (-A) (-A) (Using equation (1))
=(1)4A4=A4
i.e,(A4)T=(A4)
A4 is symmetric.
Question 5. If A=[1111] then A16 =
  1.    [02562560]
  2.    [25600256]
  3.    [−1600−16]
  4.    [016160]
 Discuss Question
Answer: Option B. -> [25600256]
:
B
A2=[0220],A4=[4004]A8=[160016],A16=[25600256]
Question 6. All the elements in a matrix A are complex numbers with imaginary parts not equal to zero. If A is the conjugate of the matrix A, aij is the general element of matrix A, then what is the general element of the matrix, A+A2.
  1.    2lm(aij)
  2.    lm(aij)
  3.    Re(aij)
  4.    Re(aij)2 
 Discuss Question
Answer: Option C. -> Re(aij)
:
C
By taking complex conjugate of a matrix we reverse the sign of imaginary parts of all the elements in the original matrix. i.e., if the element in A is x + iy, then the corresponding element in Ais x - iy.
So when A and Ais added the imaginary parts cancel out and the sum becomes 2 times the real part of element in A.
i.e., since (aij) is general element in A, the general element in A+Abecomes 2Re(aij).
General element in A+A2=Re(aij).
Question 7. If A=[1111] then A16 =
  1.    [02562560]
  2.    [25600256]
  3.    [−1600−16]
  4.    [016160]
 Discuss Question
Answer: Option B. -> [25600256]
:
B
A2=[0220],A4=[4004]A8=[160016],A16=[25600256]
Question 8. 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
A3×4A4×3Now A'B defined
Bis 3×p
Again B3×pA4×3 defined p=4
B is 3×4.
Question 9. If A is a non-diagonal involutory matrix, then
  1.    A - I = 0
  2.    A + I = 0
  3.    A - I is non zero singular
  4.    none of these
 Discuss Question
Answer: Option C. -> A - I is non zero singular
:
C
A2=IA2I=0
(A+I)(A-I)=0
either |A+I|=0 or
|AI|=0
If |AI|0, then (A+I)(AI)=0A+I=0 which is not so
|AI| and AI0.
Question 10. A=[aij]n×n and aij=i2j2 then A is necessarily
  1.    a unit matrix
  2.    symmetric matrix
  3.    skew symmetric matrix
  4.    zero matrix
 Discuss Question
Answer: Option C. -> skew symmetric matrix
:
C
aji=j2i2=(i2j2)=aij

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