NCERT Solutions Class 12

NCERT Solutions Class 12

Matrices : NCERT Solutions – Class 12 Maths (Ex 1)

Exercise 3.1

1. In the matrix A = , write:

(i) The order of the matrix.

(ii) The number of elements.

(iii) Write the elements 

Ans. (i) There are 3 horizontal lines (rows) and 4 vertical lines (columns) in the given matrix A.

Therefore, Order of the matrix is 3 x 4.

(ii) The number of elements in the matrix A is 3 x 4 = 12.

(iii)  Element in first row and third column = 19

 Element in second row and first column = 35

 Element in third row and third column = 

 Element in second row and fourth column = 12

 Element in second row and third column = 


 

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

Ans. Since, a matrix having  element is of order 

(i) Therefore, there are 8 possible matrices having 24 elements of orders 1 x 24, 2 x 12, 3 x 8, 4 x 6, 24 x 1, 12 x 2, 8 x 3, 6 x 4.

(ii) Prime number 13 = 1 x 13 and 13 x 1

Therefore, there are 2 possible matrices of order 1 x 13 (Row matrix) and 13 x 1 (Column matrix).


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

Ans. Since, a matrix having  element is of order 

(i) Therefore, there are 6 possible matrices having 18 elements of orders 1 x 18, 2 x 9, 3 x 6, 18 x 1, 9 x 2, 6 x 3.

(ii) Prime number 5 = 1 x 5 and 5 x 1

Therefore, there are 2 possible matrices of order 1 x 5 (Row matrix) and 5 x 1 (Column matrix).


 

4. Construct a 2 x 2 matrix A =  whose elements are given by:

(i) 

(ii) 

(iii)  

Ans. (i) Given:   ……….(i)

Putting  in eq. (i) 

Putting  in eq. (i) 

Putting  in eq. (i) 

Putting  in eq. (i) 

 A2 x 2 = 

(ii) Given: ……….(i)

Putting  in eq. (i) 

Putting  in eq. (i) 

Putting  in eq. (i) 

Putting  in eq. (i) 

 A2 x 2 = 

(iii) Given:  ……….(i)

Putting  in eq. (i) 

Putting  in eq. (i) 

Putting  in eq. (i) 

Putting  in eq. (i) 

  A2 x 2 = 


 

5. Construct a 3 x 4 matrix, whose elements are given by:

(i) 

(ii)  

Ans. (i) Given:  ……….(i)

Putting  in eq. (i) 

Putting  in eq. (i) 

Putting  in eq. (i) 

Putting  in eq. (i) 

Putting  in eq. (i) 

Putting  in eq. (i) 

Putting  in eq. (i) 

Putting  in eq. (i) 

Putting  in eq. (i) 

Putting  in eq. (i) 

Putting  in eq. (i) 

Putting  in eq. (i) 

  A3 x 4 = 

(ii) Given:  ……….(i)

Putting  in eq. (i) 

Putting  in eq. (i) 

Putting  in eq. (i) 

Putting  in eq. (i) 

Putting  in eq. (i) 

Putting  in eq. (i) 

Putting  in eq. (i) 

Putting  in eq. (i) 

Putting  in eq. (i) 

Putting  in eq. (i) 

Putting  in eq. (i) 

Putting  in eq. (i) 

  A3 x 4 = 


 

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

(i) 

(ii) 

(iii)  

Ans. (i)Given:  

By definition of Equal matrices, 

(ii) 

Equating corresponding entries,  ……….(i)

    ……….(ii)

And    [From eq. (i), ]

 

 

 

  or 

Putting these values of  in eq. (i), we have  and 

  

(iii)  Given:  

Equating corresponding entries,  ……….(i)

  ………. (ii)

And ……….(iii)

Eq. (i) – Eq. (ii) =  9 – 5 = 4

Eq. (i) – Eq. (iii) =  9 – 7 = 2

Putting values of  and  in eq. (i),

   

  


 

7. Find the values of  and  from the equation .

Ans. Equating corresponding entries,

 ……….(i)

 ……….(ii)

 ……….(iii)

 ……….(iv)

Eq. (i) – Eq. (ii) = 

 

Putting  in eq. (i),  

    

Putting  in eq. (iii), 

    

Putting  in eq. (iv), 

    

  


8. A =  is a square matrix if:

(A)  (B)  (C)  (D) None of these

Ans. By definition of square matrix , option (C) is correct.


 

9. Which of the given values of  and  make the following pairs of matrices equal:

 

(A) 

(B) Not possible to find

(C) 

(D)  

Ans. Equating corresponding sides,

   

And  

Also  

And 

Since, values of  are not equal, therefore, no values of  and  exist to make the two matrices equal.

Therefore, option (B) is correct.


 

10. The number of all possible matrices of order 3 x 3 with each entry 0 or 1 is:

(A) 27

(B) 18

(C) 81

(D) 512

Ans. Since, general matrix of order 3 x 3 is 

This matrix has 9 elements.

The number of choices for  is 2 (as 0 or 1 can be used)

Similarly, the number of choices for each other element is 2.

Therefore, total possible arrangements (matrices) =  times = 

Therefore, option (D) is correct.

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