WORKSHEET-16

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MATHEMATICS (Book-I) Ex #. 3.1, 3.2, 3.4

Worksheet-16
USE THIS SPACE FOR
 3  2 SCRATCH WORK
   
1. If U   2 3 4 , X   0 2 3 , V   2  , Y   2 
1  4
   
then UV  XY =
A.  4 B.  20
C. 16 D. 20
 1 2
2. The matrices C and D are given by C    and
 1 6 
 3 2 
D  . Find the matrix X such that
1 4 
CX  D  I . Where I is the identity matrix:
1  10 18  1  10 18 
A.   B.  
8  3 1  8 3 1 
1  10 18  1  10 18 
C.   D.  
8  3 1  8 3 1 
1 
4 k
 4 2 
3. For Matrices A   , B    find k that
 0 3 0 1
 
 3
makes AB identity:
2 1
A. B.
3 4
5 1
C. D.
6 6

1 a   2 1  1 5 
4. If    , values of a and b are:
 b 1   1 2   3 0 
A. 2,5 B. 1, 4
C. 2,5 D. 3, 2
 p 2  q 2  10 0 
5. If A   , B   ,C    and
 1 q  1 p   0 10 
AB  C then A 
A. 5 B. 10
C. 0 D. 2

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MATHEMATICS (Book-I) Ex #. 3.1, 3.2, 3.4

USE THIS SPACE FOR

1 2  SCRATCH WORK
6. If A    and A2  5 A1  kI 2  0, then:
3 4 
A. k  17 B. k  17
C. k  11 D. k  11
1 0
If A =  and A  A1  kI then p  ? and k  ?
P 
7.
0
A. +1, +2 B. +2, -2
1 3i
C. 2 D. +2, +1

 x
8. Find value of x , if  x 5  x      3 :
1
A. 1,0 B. 1, 2
C. 8,10 D. 2,8
9. Scalar matrix is:
1 0 0 0
A.   B.  
0 2 0 k 

2 0
C.   D. Both (B) and (C)
0 2
 r cos  0  sin    cos  0 sin  
10.  0 r 0   0 1 0  

 r sin  0 cos    r sin  0 r cos  

A. I 3 B. rI 3
C. O D. None of these
 1 2 
  1 3
If A   1 4  and B  
2 1
11. which of the
 2 1  

following is true:

 AB  B. B  AdjB   B I3
t
A.  Bt At

 AB 
1
C.  B 1 A1 D. All of these

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MATHEMATICS (Book-I) Ex #. 3.1, 3.2, 3.4

USE THIS SPACE FOR

12. A = [aij]mn is called diagonal Matrix if: SCRATCH WORK
I. aij  0, i  j
II. At least one aij  0, i  j
III. m  n
A. I and II only B. I and III only
C. II and III only D. I, II & III
13. A = [aij]23 , B = [bij]34 , C = [aij] with real entries and
 AB  C is defined then order of C is:
A. 2  4 B. 2  7
C. 4  7 D. 7  2
14. Adjoint of a square matrix is defined by:
A. Matrix of cofactors
B. Transpose of matrix of cofactors
C. Inverse of matrix of minors
D. Inverse of matrix of cofactors
1 0 
15. Inverse of matrix   is:
 z 1
1 0  1  z 0
A.   z 1 z 
B.
0 1 
 1 0  1  z 
C.   D.  
 z 1  0 1 
1 2 0 
16. If A    then AA 
t

 3 1 4 
A. O B. I
C. At A D. None of these
2 0
17. If A    and A  kA, then:
4

 0 2 
A. k  8 B. k  2
C. k  3 D. k  4
1 0 2   3 2 4
  1  1  , then:
18. If A  0 2 1  and A   1 1
0 1 1   2 1 k 
A. k  1 B. k  2
C. k  2 D. k  3

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MATHEMATICS (Book-I) Ex #. 3.1, 3.2, 3.4

USE THIS SPACE FOR

If A and B are skew-symmetric matrices then  AB 
t
19. SCRATCH WORK
is equal to:
A. At Bt B.  AB
C. AB D. BA
20. If A is a square matrix, then which one is correct:
A. A  A is symmetric B. A  A is symmetric
t t

C. A  A is skew symmetric D. A  A is hermitian

t t

21. If A is skew Hermitian Matrix then which of the

following is not skew Hermitian matrix:
A. A2 B. A5
C. A3 D. A7
22. Which of the following is skew symmetric matrix:
0 1 2  0 1 2 
A. 1 0 3  B.  1 0 3
   
 2 3 0   2 3 0 

 0 0 0  6 1 2 
C.  1 0 0  D.  1 6 3
   
 2 3 0   2 3 6 

1 2 
23. Rank of matrix   is:
3 0 
A. 0 B. 2
C. 1 D. 3
24. Which of the following is a skew Hermitian matrix:
3  i i   0 2  3i 
A.   B.  
 2i 1  i   2  3i 0 
 1 1 i  3i i  1
C.   D.  
1  i 2  i  1 5 
25. If A is a square matrix, then which of the following is
not symmetric matrix:
A. A  At B. AAt
C. At A D. A  At

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MATHEMATICS (Book-I) Ex #. 3.1, 3.2, 3.4

USE THIS SPACE FOR

 x x  2
If B    is a symmetric matrix, then x 
26. SCRATCH WORK
 2 x  3 x  1 
A. 3 B. 5

C. 2 D. 4
27. For A   aij  , if aij  0, i  j then the matrix is:
nn
A. Symmetric Matrix B. Hermitian Matrix
C. Lower Triangular D. Upper Triangular
1 2 1 3 
28. Rank of matrix  4 6 7 8  is:
 
0 4 10 8

A. 1 B. 2
C. 4 D. 3
29. Which one is in reduced echelon form only?
1 2 
A.  2 1 0
t
B.  
5 6 
1 2 1
 1 1 0 0 
C. 0 2 3  D.  
  0 0 1 3
 0 0 1 

1 1 1 3
2 0 0 2 
30. Rank of the matrix  is:
0 2 0 2
 
0 0 2 2
A. One B. Two
C. Three D. Four

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MATHEMATICS (Book-I) Ex #. 3.1, 3.2, 3.4

ANSWER KEY (Worksheet-16)  1 a   2 1  1 5 

4. (D)    
1 B 11 A 21 A  b 1   1 2   3 0 
2 A 12 D 22 B  2  a 1  2a   1 5 
3 D 13 C 23 B   
 2b  1 b  2   3 0 
4 D 14 B 24 B
 2  a  1 2b  1  3
5 B 15 C 25 D
 a  3 b2
6 B 16 D 26 B
 p 2
7 A 17 A 27 C 5. (B) A   
8 B 18 C 28 D  1 q
9 C 19 D 29 D  A  pq  2 ….. (i)
10 B 20 A 30 C Now AB  C
 p 2   q 2  10 0 
ANSWERS EXPLAINED    
 1 q   1 p   0 10 
 3  2
Comparing 1,1 th element
 
1. (B) UV  XY   2 3 4  2   0 2 3  2 
    pq  2  10
1  4
    Put in (i)
  6  6  4   0  4  12 A  pq  2  10
  4  16   20 1 2 
6. (B) Here A   
 1 2  3 2  3 4 
2. (A) Here C   , D   
 1 6  1 4  1 2  1 2   1  6 2  8 
A2     
CX  D  I  CX  I  D 3 4  3 4  3  12 6  16 
 1 0   3 2    2 2   7 10 
      
 0 1   1 4   1 3  15 22 
 2 2  1  6 2   2 2  1  4 2
 X  C 1      A1 
1
AdjA 
 1 3  8  1 1   1 3  A 4  6 3 1 
1  12  2 12  6  1  10 18 
     1  4 2 
8  2  1 2  3  8  3 1 A1 
2  3 1 
3. (D) By given AB  I
1  By given A2  5 A1  kI 2  0
 k  7 10    4 2  1 0  0 0 
 4 2  4 1 0    k  
      1  0 0 
 0 3 0 1  0 1 15 22  2  3 1  0
   10 5
 3  7 10   k 0  0 0 
   15 5    
Multiplying 1st row of first matrix by 2nd
15 22     0 k  0 0 
2  2 2
column of second matrix  4k 
3 Comparing a11 on both sides
Comparing 1, 2  th element on both sides 7  10  k  0
 k  17
2 1  2  1
4k  0 k    1 0 
3 4 3  6 7. (A) Here A   
0 p 

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MATHEMATICS (Book-I) Ex #. 3.1, 3.2, 3.4

1 1  p 0  r cos 2   r sin 2  0  0  0 r cos  sin   r cos  sin  

1
A  AdjA    
p  0 1   000 0r 0 000 
A  r cos  sin   r cos  sin  000 r sin 2   r cos 2  
 
By given A  A1  kI  r 0 0 1 0 0 
1 0   
 0 r 0  r 0 1 0   rI 3
1 0     k 0
   1   0 0 r  0 0 1 
0 p  0 p   0 k 
 
11. (A) Since the matrix A is not square so its
2 0  inverse does not exist so option (C)
    k 0  involve A1 , cannot be the answer.
0 p    0 k 
1
 p  The product AB and Bt At exists.
So (A) is the required option.
1
 k  2 p   k _____ (i) Moreover B  AdjB   B I 2
p
Put k  2 in (i) So (B) is not true.
1 12. (D) If A   aij  is a diagonal matrix then
p 2 mn
p it must square i.e. m  n and non-
 p2  2 p  1  0 diagonal elements all zero but diagonal
elements, not all zero.
  p  1  0  p  1
2
Hence all I, II and III are true.
Hence k  2 p  1 13. (C) Order of A is 2  3
 x Order of B is 3  4
8. (B)  x 5  x      3 Then the order of AB is 2  4
1
Now if  AB  C is defined then C must
  x  5  x    3
2

have order 4  n
 x2  5  x  3 Option (C) is of this type.
x2  x  2  0 14. (B) Adjoint is defined by “Transpose of the
 x2  x  2  0 matrix of cofactors”.
b  b 2  4ac 1
x 15. (C) Since A1  AdjA
2a A
1
1  1  4 1 2  1  7i 1 0 1 1 0  1 0
x        
2 1 2  z 1 1 0  z 1  z 1 
This equation is satisfied by x  2 and 1 3
 1 2 0 
x  1 only. 16. (D) AA  
t
 2 1
9. (C) A square matrix is a scalar matrix if  3 1 4   0 4 
aij  0 i  j and aij  k  0 i  j 
1  4  0 3  2  0 
i.e. non-diagonal elements are zero and  
diagonal elements are equal to a single 3  2  0 9  1  16 
non-zero scalar. 5 1 
 r cos  0  sin    cos  0 sin      O and  I
1 26 
10. (B)   0 0 
 0 r 0  1
Also order of AAt is 2  2
 r sin  0 cos    r sin  0 r cos   But order of At A is 3  3

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MATHEMATICS (Book-I) Ex #. 3.1, 3.2, 3.4

Hence AAt  At A Number of non-zero rows in reduced

So (D) is required option. echelon form or echelon form is called
17. (A) For a diagonal matrix rank of matrix
a 0  a n 0 So Rank  2
If A     A n
   Note:
0 a 0 an  For a 2  2 non-zero matrix
2 0  24 0  Rank  1 , if A  0
So if A   4

2   4
then A
0 0 2  and Rank  2 , if A  0
16 0 2 0 24. (B) In a skew Hermitian matrix
 A4     8   kA
0 16  0 2 aij  0 or i (pure imaginary)  i  j
Hence k  8 and aij  a ji i  j
18. (C) AA1  I i.e. diagonal elements are zero or pure
1 0 2  3 2 4  1 0 0  imaginary and elements opposite to main
 0 2 1   1 1 1   0 1 0 diagonal are the negative conjugate of
each other.
0 1 1   2 1 k  0 0 1 
Only (B) option satisfy this condition.
Comparing 1,3 rd element 25. (D) For a square matrix A
1 4  01   2 k   0 AAt , At A, A  At all are symmetric
4  2 k  0  k  2 But A  At is skew-symmetric
26. (B) Since B is symmetric
19. (D) If A and B are skew symmetric then
So a12  a21
At   A and Bt  B
Now  2x  3  x  2  x  5
27. (C) aij  0  i  j
 AB   Bt At   B   A  BA
t

 Elements above main diagonal are zero

20. (A) For a square matrix A
then the matrix is called lower triangular.
A  At is always symmetric and
1 2 1 3 
A  At is always skew-symmetric.
28. (D)  4 6 7 8 
21. (A) If A is skew-Hermitian  
 Hermitian, if n is even 0 4 10 8
Then An  
 Skew Hermitian, if n is odd  1 2 1 3 
 0 2 11 4  R  4 R
So A2 is not skew-Hermitian. R~   2   1
22. (B) A square matrix is skew-symmetric. If  0 4 10 8 
aij  0  i  j and aij  a ji  i  j  1 2 1 3 
 0 2 11 4  R  2 R
i.e. each diagonal element is zero and R~   3   2
elements opposite to main diagonal are  0 0 12 0 
negative of each other. 1 2 1 3  1
1 2   1 2   11  2 2
R
23. (B)   R  R2   3 R1 R  0 1  2
3 0  0 6  ~  2  1
0 0 R
1 2  1  1 0  12 3
R   R2
0 1  6 The number of non-zero rows in echelon
form of matrix is 3, so rank  3

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MATHEMATICS (Book-I) Ex #. 3.1, 3.2, 3.4

29. (D) A matrix will be in reduced echelon

form if
(i) “the number of leading zeros in a row
is greater than the number of such zero’s
in the preceding row.”
(ii) “Each element above or below the
leading element must be zero.”
Only (D) option satisfy above-mentioned
conditions.
30. (C) We reduce the matrix in echelon form
1
1 1 1 3  1 1 1 3 2 R2
 2 0 0 2  1 0 0 1
  R  1R
 0 2 0 2  0 1 0 1 2 3
   
 0 0 2 2  0 0 1 1 1 R4
2
1 1 1 3 
0 1 1 2 
 R R R
0 1 0 1  2 1

 
0 0 1 1 
1 1 1 3 
0 1 1 2 
 R   1 R
0 1 0 1  2

 
0 0 1 1 
1 1 1 3 
0 1 1 2 
 R R R
0 0 1 1 3 2

 
0 0 1 1 
1 1 1 3 
0 1 1 2 
 R R R
0 0 1 1 4 3

 
0 0 0 0 
It is clear from the last step that the
number of non-zero rows in echelon form
is 3
So Rank  3

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