Chapter 3 : Matrices

Class 12 Chapter 3 : Matrices  Exercise 3.1

1. In the matrix A , write: (i) The order of the matrix, (ii) The number of elements, (iii) Write the elements  .

Solution:  (i) The order of the matrix is 3 × 4 .

(ii) The number of elements are 3 × 4 = 12 .

(iii)  Given, the matrix

Here,  ,,  , ,

2. If a matrix has 24 elements, what are the possible orders it can have? What, if it has 13 elements?

Solution:  We know that if a matrix is of order  , it has  elements.

Thus, all possible ordered pairs are (1, 24), (24, 1), (2, 12), (12, 2) ,(3, 8) , (8, 3) , (4 , 6) , (6 , 4)

Hence, possible orders are 1 × 24, 24 ×1, 2 × 12, 12 × 2 , 3 × 8 , 8 × 3 , 4 × 6 , 6 × 4 .

Thus, all possible ordered pairs are (1,13) , (13,1) .

Hence, possible orders are 1 × 13 , 13 × 1 .

3. If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements?

Solution:  We know that if a matrix is of order  , it has  elements.

Thus, all possible ordered pairs are (1, 18) , (18, 1) , (2, 9) , (9, 2) , (3 , 6) , (6 , 3)

Hence, possible orders are 1 × 18 , 18 ×1 , 2 × 9 , 9 × 2 , 3 × 6 , 6 × 3

Thus, all possible ordered pairs are (1 , 5) , (5 , 1) .

Hence, possible orders are 1 × 5, 5 × 1 .

4. Construct a 2 × 2 matrix, whose elements are given by :

(i)

(ii)

(iii) 

Solution:  (i) We have, 

 We construct 2 × 2 matrix  

Now ,

;   ;    ;

Hence, the required matrix is given by

(ii) We have, 

 We construct 2 × 2 matrix

 Now,

, , and

Hence, the required matrix is given by

(iii) We have 

We construct 2 × 2 matrix

  ;  ; ;

Hence, the required matrix is given by

5. Construct a 3 × 4 matrix, whose elements are given by:  (i)      (ii)  

Solution:   We construct 3 × 4 matrix

 (i)  We have ,

;    ; ;   ;   ; ;

;   ; ;   ; and

 Hence, the required matrix is given by

(ii) We have,  

We construct 3 × 4 matrix

Now,  ,  ,  , ,  ,  ,  , ,  ,  , ,

Hence, the required matrix is given by

6. Find the values of  and  from the following equations :

(i)     (ii)    (iii)               

Solution:  (i) We have,    

By equality of two matrices , equating the corresponding elements , we get

     , , 

(ii) We have,   

By equality of two matrices , equating the corresponding elements , we get

 Therefore,

 …….. (i)

 

  and

   [From (i)]

 

Or 

Putting  in (i) , we get

Putting  in (i) , we get 

Therefore,  or 4 ,  or 4  and

 (iii) We have,         

By equality of two matrices , equating the corresponding elements , we get

       ………… (i)

  ………… (ii)

  …………….(iii)

 Putting  and   in (i) , we get

Therefore, ,  and
7. Find the value of a, b, c and d from the equations :            

Solution : We have,   

  By equality of two matrices , equating the corresponding elements , we get

    

 

 

 

Putting  in (i) , we get

Putting  in (iii) , we get 

Putting  in (iv) , we get

                     
8.   is a square matrix, if
(A)     (B)   (C)    (D) None of these

Solution:  Since, the number of rows are equal to the number of columns in a square matrix .

        i.e.,

The correct Answer  (C)  .

9. Which of the given values of x and y make the following pair of matrices equal

         

(A)    (B)  Not possible to find   

 (C)        (D) 

Solution:  We have ,       

By equality of two matrices , equating the corresponding elements , we get

  

 

The correct answer (C)
10. The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is:
(A) 27   (B) 18    (C) 81    (D) 512

Solution:  We know that the number of elements are 3 × 3 = 9 .

 Therefore , the number of all possible matrices of order 3 × 3 with each entry 0 or 1 is  .

Class 12 Chapter 3 : Matrices  Exercise 3.2

1. Let  ,    , . Find each of the following :

(i)      (ii)     (iii)     (iv)     (v)

Solution: The matrix are  ,    ,  

(i) We have,          

(ii) We have,    

(iii) We have,

 (iv) We have,    

(v) We have,

2. Compute following :

(i)     (ii)     

(iii)      (iv)   

Solution : (i) We have,   

(ii) We have,    

(iii) We have,   

 (iv) We have,

[ Putting  ]

3. Compute the indicate products .

(i)        (ii)      (iii)   

(iv)        (v)    (vi)   

Solution:  (i) We have,

(ii)We have,  

 (iii) We have,  

(iv) We have,

(v) We have,

(vi) We have, 

4. If A  ,   ,   , then compute  and  . Also , verify that   .

5. If and  , then compute  .

6. Simplify :    

7. Find  and  , if  (i) and  

(ii)  and

8.   Find X , if   

9. Find  and  , if  

10. Solve the equation for  and   , if 

11. If  , find the values of  and  .

12.Given   , find the values of  and  .

13. If   , Show that  .

14. Show that  :

(i)  

(ii)   

15. Find    ,  .

16. If  , Prove that  .

17.  If  and  , find  so that  .

18. If  and  is the identity matrix of order 2 , show that  .

19.  A trust fund has Rs 30,000 that must be invested in two different types of bonds. The first bond pays 5% interest per year, and the second bond pays 7% interest per year. Using matrix multiplication, determine how to divide Rs 30,000 among the two types of bonds. If the trust fund must obtain an annual total interest of: (a) Rs 1800 (b) Rs 2000

20. The bookshop of a particular school has 10 dozen chemistry books, 8 dozen physics books, 10 dozen economics books. Their selling prices are Rs 80, Rs 60 and Rs 40 each respectively. Find the total amount the bookshop will receive from selling all the books using matrix algebra.
Assume X, Y, Z, W and P are matrices of order 2 × n, 3 × k, 2 × p, n × 3 and p × k, respectively. Choose the correct answer in Exercises 21 and 22.
21. The restriction on n, k and p so that PY + WY will be defined are:
(A) k = 3, p = n (B) k is arbitrary, p = 2   (C) p is arbitrary, k = 3 (D) k = 2, p = 3

22. If , then the order of the matrix  is:
(A)   (B)  (C)   (D) 

Class 12 Chapter 3 : Matrices  Exercise 3.3

1. Find the transpose of each of the following matrices :

(i)        (ii)     (iii)    

2. If   and   , then verify that

 (i)  

(ii) '   

3. If   and B , then verify that :

 (i)  

(ii) '   

4. If  and B  , then find  .

5. For the matrices A and B , verify that  , where

(i) A  ,      (ii)  ,  

6. If  (i)  , then verify that  .

(ii)   , then verify that  .

7. (i) Show that the matrix  is a symmetric matrix .

(ii) Show that the matrix  is a skew symmetric matrix .

8. For the matrix   , verify that

(i)  is a symmetric matrix .

(ii)  is a skew symmetric matrix .

9.Find and   , when  

10. Express the following matrices as the sum of a symmetric and a skew symmetric matrix .

(i)       (ii)     (iii)    (iv)  

Choose the correct answer in the Exercises 11 and 12 .

11.  If  are symmetric matrices of same order , then  is a

(A) Skew symmetric matrix    (B) Symmetric matrix      (C) Zero matrix     (D) Identity matrix

12. If  ,then  , if the value of  is

(A)          (B)       (C)      (D)   

Class 12 Chapter 3 : Matrices  Exercise 3.4

Using elementary transformations, find the inverse of each of the matrices , if it exists in Exercises 1 to 17 .

1.    2.   3.   4.   5.   6.

7.    8.   9.   10.     11.    12.

13.     14.    15.    16.       17.  

18. Matrices A and B will be inverse of each other only if 

(A) (B)   (C)    (D)

Class 12 Chapter 3 : Matrices Miscellaneous Exercises

1. Let   , show that  , where  is the identity matrix of order 2 and  .

2. If    , prove that   .

3. If   , then prove that  , where  is any positive integer .

4. If A and B are symmetric matrices , prove that  is a skew symmetric matrix .

5. Show that the matrix B’AB is a symmetric or skew symmetric according as A is symmetric or skew symmetric .

6. Find the values of   if the matrix   satisfy the equation '  .

7. For what value of :   ?

8. If , show that  .

9. Find   if   

10. A manufacturer produces three products  which he sells in two markets. Annual sales are indicated below:

Market                Products
   I     10,000          2,000          18,000
  II       6,000        20,000            8,000
(a) If unit sale prices of  and  are Rs 2.50, Rs 1.50 and Rs 1.00, respectively, find the total revenue in each market with the help of matrix algebra.
(b) If the unit costs of the above three commodities are Rs 2.00, Rs 1.00 and 50 paise respectively. Find the gross profit.
11. Find the matrix  so that   .
12. If A and B are square matrices of the same order such that AB=BA , then prove by induction that  . Further, prove that (AB)n=AnBn for all n∈N.
Choose the correct answer in the following questions:
13. If   is such that   , then
(A)     (B)
(C)         (D)
14. If the matrix A is both symmetric and skew symmetric, then
(A) A is a diagonal matrix        (B) A is a zero matrix
(C) A is a square matrix           (D) None of these
15. If  is square matrix such that , then  is equal to
(A)        (B)         (C)             (D)