XII MATHS ASSIGNMENT

MATRICES & DETERMINANTS

Q.1 If A and B are matrices of order 3  n and m  5 respectively, then find the order of
matrix 5 A  3B , given that it is defined.
Ans. m  3, n  5 . Hence order of the matrix 5 A  3B is 3  5 .

Q.2 Find the value of A 2 , where A is a 2  2 matrix whose elements are given by
1 if i  j
aij  
0 if i  j
0 1   0 1   0 1   0  1 0  0  1 0 
Ans. A   A2     .
1 0   1 0  1 0   0  0 1  0   0 1 

Q.3 Given that A is a square matrix of order 3  3 and A  4 . Find adj A .

adj A   4 
31
Ans.  16 .

Q.4 Let A   aij  be a square matrix of order 3  3 and A  7 . Find the value of
a11 A21  a12 A22  a13 A23 where Aij is the cofactor of element aij .
Ans. If elements of a row (or column) are multiplied with cofactors of any other row (or
column), then their sum is zero.
So, a11 A21  a12 A22  a13 A23  0 .

Q.5 Find the value of x  y , if

 1 3   y 0  5 6 
2   .
 0 x   1 2  1 8 
Ans. x y  0.

2 0 1
Q.6 If A   2
 1 3  , then find  A2  5 A  .
 1 1 0 
 5 1 3 
Ans.  1 7 10 

 5 4 12 

Q.7 If A is a square matrix of order 3, such that A  adj A  10 I, then adj A is equal to
(a) 1 (b) 10 (c) 100 (d) 101
Ans. No Answer

Q.8 If A is a 3  3 matrix such that A  8 , then 3A equals.

(a) 8 (b) 24 (c) 72 (d) 216
Ans. No Answer

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1 0   1 1
Q.9 If A  B    and A  2 B   , then A = _________ .
1 1   0 1
Ans. No Answer
 2  0 
Q. 10 Find the order of matrix  1 1  .
  
 4   3 
'
 2  0   2 
Ans. Order is 3  3 as  1 1    1  0 1 5 .
    
 4   3   4 

Q. 11 Given a matrix A of order 3  3 . If A  3 , then find A. Adj A .

As A  Adj A  A   3  27 .
3 3
Ans.

 cos  sin  
Q. 12 If matrix A    , then find the value of  for which A is an identity
  sin  cos  
matrix.
Ans.  cos   1,sin   0    0
5  x x  1
Q. 13 Find the value of x , for which the matrix  is singular.
 2 4 
Ans.  20  4 x  2 x  2  0  6 x  18  x  3 .

Q.14 If matrix A  1 3 0 , then write AA ' , where A ' is transpose of matrix A.

 1
Ans. AA '  1 3 01 3 0 '  1 3 0  3  1  9  0  10 .
 0 

 0 2  0 2a 
Q.15 If matrix A    and matrix B    , find the value of k , a and b, if kA  B
1 4 b 24 
.
Ans. k  6  a  6, b  6 .

 2  1 10 
Q.16 If x    y      , find x .
 3  1  5 
Ans. 5 x  15  x  3

 3 2
Q.17 If matrix A   1
 , then find A .
 1 4 
1
Ans.  A1  .
14

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a b 
Q.18 Find the inverse of the matrix A   .
 c 1  bc 
 a 
1  bc  1  bc   1 
1 b   b  1
Ans.  A 1   a a  A    adj A  
1    A 
 c a   c a

x  3y y   4 1
Q.19 If   , find the values of x and y .
 7z 4   0 4 
Ans.  x  3 y  4; y  1;7  x  0  x  7; y  1

Q.20 Write the possible orders of a matrix having 18 elements.

Ans. No Answer

Q.21 Given matrices A and B of orders m 4 and 3  n respectively. If 2 A  5B is defined

then write its order.
Ans. No Answer

x  2 3
Q.22 If  3 , find the integral value (s) of x.
3x 2 x
Ans. No Answer

a11 a12 a13

Q.23 Given determinant   a21 a22 a23 and Aij represent cofactor of an element aij ,
a31 a32 a33
then find value of a11  A11  a12  A12  a13  A13 , if   15 .
Ans. No Answer

 1 0
Q.24 If A    , find k so that A  8 A  kI .
2

 1 7 
Ans. No Answer

2 j  i
2

Q.25 Given a matrix A   aij  of order 3  3 whose elements aij are given by aij  ,
i j
then find the element a32 of matrix A.
 4  3
2
1
Ans. a32   .
3 2 5

Q.26 Given a skew symmetric matrix A and a matrix B such that B ' AB is defined, then
find B ' AB .
Ans. Hence, B ' AB is a skew symmetric matrix.

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Q.27 If A and B are square matrix of order 3 such that A  1 and B  3 , then find the
value of 3AB .
Ans. As AB is of order 3.
and 3 AB  33 AB  27 A B  27 1 3  81 .

2
4 1 3 2 x 3
Q.28 If  
, then find the value of x.
2 1 1 x 2 1
Ans.  4  2 x  8  2 x  12  x  6 .

cos   sin  
Q.29 If A    , find the value (s) of  such that A ' A  I 2 .
 sin  cos  

Ans.   .
3

3 2  2 1
Q.30 Solve the matrix equation   A  , using the concept of inverse of a
7 5  0 4 
matrix.
 10  0 5  8  10 13
Ans.   .
 14  0 7  12   14 19 

1 3 3 
Q.31 If A  1 4 3  , then verify that A  adj A  A I . Also, find A1 .
1 3 4 
 7 3 3
adj A 
Ans.  1
A    1 1 0  .
A
 1 0 1

Q.32 If A is any non – zero matrix of order n, then find A. Adj A , in terms of A and
identity matrix I.
Ans. A  Adj A  A I .
k 3 4 3
Q.33 For what value of k  N ,  ?
4 k 0 1
Ans.  k  4 N

a b c a d g
Q.34 If b e f  p , then find the value of b e h , if p  20 .
g h i c f i

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a b c
Ans.   d e f  p  20 .
g h i
Q.35 For the matrixes A and B, if multiplications is defined and AB  A and BA  A , then
find B 2 .
Ans.  BA  B 2  B  B 2 .

 1 1
Q.36 If matrix A    and A  kA , then find the value of k.
2

 1 1 
Ans.  k  2.

 2 3 1
Q.37 If A   1 2 2  , find A1 and hence show that how we can use A1 to solve the


 3 1 1
system of equations 2 x  y  3 z  13 ; 3 x  2 y  z  4, x  2 y  z  8 .
    A1  i.e. we take transform of A1
1 T
Ans. Now we have A1 [from (i)] and use AT
obtained and substitute in (iii) to get X and then x, y, z.

Q.38 Given a matrix A   aij  of order 2  3 , where elements are given by aij  i 2  j  3 ,
then find the element at the first row and third column.
Ans. No Answer.

 3 0 2b 
Q.39 If matrix 0 1 4  is symmetric then find the value of b.
5 a 5 
Ans. No Answer.

2x 5 6 2
Q.30 If  , then find the value of x   0  .
8 x 7 3
Ans. No Answer.

a11 a12 a13

Q.41 Given   a21 a22 a23 , such that   9 . Find the value of a11  A31  a12 A32  a13 A33
a31 a32 a33
where Aij represents cofactor of element aij of a determinant.
Ans. No Answer.

1 1
Q.42 If A    and A  2 A , then find the value of k.
100 k

1 1
Ans. Now comparing with A100  2 k A , we get k  90 .

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Q.43 Show that AA ' and A ' A are both symmetric matrices for any matrix A.
Ans.  A ' A '  A '  A ' '  A ' A . Hence, symmetric.
3 x 3 x x 3
Q.44 Find the solution of the determinant 6 3 1 0.
3 6 3
Ans.  x  3.

 2 1 5
Q.45 If the matrix  0 3 k  is singular, then find k.
1 3 2 
27
Ans.  For k  , given matrix is singular.
5

a 0 0 
Q.46 If matrix A   0 a 0  then, find A2 .
 0 0 a 
Ans.  diage  a 2 , a 2 , a 2  .
a b c
Q.47 Find the value of the determinant 4 3 2 .
ka kb kc
Ans.  3kac  2kab  4kbc  2kab  4kbc  3kac  0 .

 2 3
Q.48 Let A    and f  x   x 2  4 x  7 . Show that f  A  0 , hence find A5 .
 1 2 
 62  56 93  0   118 93 
Ans. =   .
 31  0 62  56   31 118

3 0 0
Q.49 
If A  0 2 0  , then A is a diagonal or a scalar matrix?
0 0 4 
Ans. No Answer.

Q.50 If A is a matrix of order 3  5 and B is a matrix such that A ' B and BA ' are both
defined, then find the order of matrix B.
Ans. No Answer.
 3 1 2
Q.51 Given matrix A   4 5 9  , write the value of 3a22  4a33 .
1 3 4
Ans. No Answer.

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2 3 4
Q.52 Find the value of the determinant, 5 6 8 .
6 x 9 x 12 x
Ans. No Answer.

3 0 0 
Q.53 If A  0 2 0  , then A is a diagonal or a scalar matrix?
0 0 4 

Q.55 For a square matrix A, show that A  A ' is a symmetric matrix and A  A ' is a skew
symmetric matrix.
Ans. No Answer.

 1 0
Q.56 Solve the following matrix equation for x,  x 1    0.
 2 0 
Ans. No Answer.

 4 4 4  1 1 1 
Q.57 Determine the product  7 1 3  1 2 2  and state how it can be used to
 5 3 1  2 1 3 
solve the system of equations x  y  z  4; x  2 y  2 z  9; 2 x  y  3 z  1 .
Ans. No Answer.

1 2 2 
1
Q.58 If A   2 1 2  and AA '  I , then find x  2 y .
3
 x 2 y 
Ans. No Answer.

Q.59 Show that the all the elements on the main diagonal of a skew symmetric matrix are
zero.
Ans. No Answer.

0 a 3
Q.60 If the matrix  2 b 1 is a skew symmetric matrix, find a, b and c.
 c 1 0 
Ans. No Answer.

 0 3 1
Q.61 If matrix A   1 4 2  , then find A1 .
 1 1 1 
Ans. No Answer.

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 0 1 1 
Q.62 If B    and A    . Find the value of  such that A  B 2 .
 1 1 5 0 
Ans. No Answer.

a 0 0 
Q.63 If A   0 a 0  , then find the value of Adj A .
 0 0 a 
Ans. No Answer.

x 2 x
Q.64 Let x2 x 6  ax 4  bx 3  cx 2  dx  e , then find value of 5a  4b  3c  2d  e .
x x 6
Ans. No Answer.

Q.65 If matrix A  diag  1, 4 2 , then find 2 A  4 I .

Ans. No Answer.

 4 3
Q.66 Given matrix A    , show that A  A ' is symmetric matrix and A  A ' is a skew
 1 2 
symmetric matrix.
Ans. No Answer.

If A is a square matrix such that A2  A , show that  I  A   7 A  I .

3
Q.67
Ans. No Answer.

2 3 10 
Q.68 
If A   4 6 5  , find A1 . How we can use A1 to solve the system of equations
 6 9 20 
2 3 10 4 6 5 6 9 20
   2;    5;    4? .
x y z x y z x y z
Ans. No Answer.

 p 0 1 0 
Q.69 If A    and B    , then find the value of p for which A2  B .
 1 1 2 1
Ans. No Answer.

Q.70 If A and B are invertible matrices of the same order and A 1 , B 1 are known, then to
find  AB  , how do we use A1 and B 1 .
1

Ans. No Answer.

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e2log3 4
Q.71 Find the value of 1 .
1 log  
e
Ans. No Answer.

 2 0 1  1 
Q.72 Write the order of the product of matrices 3 1 4  1 2 2   6  and find its
 5 4 1  9 
value.
Ans. No Answer.

2 0 1
Q.73 If A  2 1 3 , find A2  5 A  4 I and hence find a matrix X such that
1 1 0
A2  5 A  4 I  X  0 .
Ans. No Answer.

2 0  3 2 2
Q.74 If A is a scalar matrix   and B is a matrix  4 1  then find A B .
0 2  
Ans. No Answer.

2
Q.75 Given a matrix A of order 3  3 and A  5 , then find Adj A .
Ans. No Answer.

 x  y x  2  8 5 
Q.76 If     find the value of  y  x  .
 2 x  y 16  1 3 y  1
Ans. No Answer.

Q.77 Using determinants find the value of k such that area of a triangle with vertices  0,3 ,
 1, 2 and  k ,0  is 4.
Ans. No Answer.

Q.78 If A and B are squares matrices of order 3 such that A  1, B  3, then find the
value of 2 AB .
Ans. -24

Q.79 If A is a skew-symmetric matrix or order 3 (odd order), then prove that det A = 0.
Ans. A  A.

Q.80 If A and B are square matrices of the same order 3, such that A  2 and AB  2 I ,
write the value of B .

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Ans. B  4.

Q.81 If A is a square matrix of order 2 and A  4, then find the value of 2 AAT , when AT
is the transpose of matrix A.
Ans. 64

Q.82 If A is any square matrix of order 3×3 such the adj A  169 and A is non-negative,
then the value of A is?
(a) 13 (b) -13 (c) 13 (d) None of these
Ans. 64

 5 6 
Q.83 If adj A    then A-3 is:
 7 4 
 5 6  4 6 1  5 6 1  4 6
(a)   (b)   (c)  (d) 
 7 4  7 5  22  7 4  22 7 5
1  5 6
Ans.  
22  7 4 

Q.84 If A is any square matrix of order 3×3 such the adj A  4, then the value of A is?
(a) 2 (b) -2 (c) 2 (d) None of these
Ans. (c) 2

Q.85 If A and B are square matrices of order 3 each, A  2 and B  3 then the value of
3AB is ______________
Ans. 162

Q.86 If A is any square matrix of order 3×3 such the A  4, then the value of A.adj A is?
(a)16 (b) 64 (c) – 16 (d) – 64
Ans. (c) – 16

Q.87 Which of the following is correct?

(a) Determinant is a square matrix.
(b) Determinant is a number associated to a square matrix.
(c) Determinant is a number associated to a matrix.
(d) None of these
Ans. (b) Determinant is a number associated to a square matrix

Q.88 If A is any square matrix of order 3×3 such the A  3, then the value of adj A is?
1
(a) 3 (b) (c) 9 (d) 27
3
Ans. (c) 9

Q.89 If A is any square matrix of order 3×3 such the A  7, then the value of adj A is?

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(a) 49 (b)343 (c) -49 (d) -343

Ans. (a) 49

Q.90 If A is any square matrix of order 3×3 such the A  5, then the value of AAT is?
(a) 5 (b) 25 (c) 125 (d) 1
Ans. (b) 25

Q.91 If A is any square matrix of order 3×3 such the adj A  25 and A is non-negative,
then the value of A is?
(a) 5 (b) -5 (c) 5 (d) None of these
Ans. (b) – 5

Q92. If A is any square matrix of order 2×2 such the A  7, then the value of adj A is?
(a) 7 (b) 49 (c) -7 (d) -49
Ans. (c) – 7

Q.93 The value of 3I3 , where I3 is the identity matrix of order 3 is:
(a) 3 (b) 9 (c) 27 (d) None of these
Ans. (c) 27

x x 3 4
Q.94 If  , write the positive value of x is:
1 x 1 2
(a) 1 (b) 2 (c) 3 (d) – 1
Ans. (b) 2

Q.95 If A is a square matrix of order 3 and 2 A  K A , then the value K is:

(a) 2 (b) 4 (c) 8 (d) None of these
Ans. (c) 8

Q.96 If A is any square matrix of order 3  3 such that A  4, then the value of A1 is:
1
(a) 4 (b) 16 (c) 2 (d)
4
1
Ans. (d)
4

Q.97 If KA  245 and A  5 for 3  3 order. Then find value of K.

K   49 
1/3
Ans.

3 2 
Q.98 A  then find A adj A without finding adj A.
1 5 
17 0 
Ans.  
 0 17 

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Q.99 If A  8 for 3  3 order then find : A  adj A 

Ans. -512

Q.100 If A  adj. A   784 for 2  2 order then find A .

Ans. 28

Q.101 If A  4, B  5 for 3  3 order, find : 7AB

Ans. -7260

 2k  3 4 5 

Q102. If the matrix  4 0 6  is skew-symmetric, then the value of k is
 5 6 2k  3
3 3 1
(a) (b)  (c) (d) None of these
2 2 2
3
Ans. (b) 
2

Q.103 For a square matrix A, A+AT is

(a) Symmetric matric (b) Skew-symmetric matrix
(c) Identity matrix (d) Void matrix
Ans. (a) Symmetric matric

Q.104 If A and B are two matrices such that AB – A and BA = B what is the value of B2.
Ans. A+B

Q.105 If A and B are square matrices of order 3 and A  5 and B  3 , then find the value
of 3AB .
Ans. 405
2 2 0
Q.106 Find the matrix A such that A  2 and adj A   2 5 1  .
 0 1 1 
 4 2 2   2 1 1 
1
Ans. A  2  adj A   1   2 2 2    1 1 1
2
 2 2 6   1 1 3 

Q.107 If A and B are matrices of the same order, then ABT  BT A is a:

(a) Skew-symmetric matrix (b) null matrix
(c) Unit matrix (d) symmetric matrix
Ans. (a) Skew-symmetric matrix

Q.108 If A is a square matrix, then AA IS A

(a) Skew-symmetric matrix (b) symmetric matrix
(c) diagonal matrix (d) none of these
Ans. (a) Skew-symmetric matrix
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Q.109 If A and B are symmetric matrices, then ABA is

(a) Symmetric matrix (b) skew-symmetrc matrix
(c) diagonal matrix (d) scalar matrix
Ans. (a) Symmetric matrix

Q.110 If A be any square matrix of order 3×3 and A  5, then find the value of adj  adj A 
Ans. 625
3
Q.111 If A   1 2 3 and B   4  find AB .
 0 
Ans. – 11
a 0 0 
Q.112 Find A  adjo int A  and adjo int A , if A   0 a 0  .
 0 0 a 
Ans. a9 , a6

Q.113 If A is a 3×3 matrix, A  0 and 3A  K A , then write the value of K.

Ans. K=27

10 0 
Q.114 For any 2×2 matrix A, if A  adjo int A     , then find A .
 0 10 
Ans. A  10.

 2 1
Q.115 If A    , find A  adjA 
 7 5
Ans. 9

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