NCERT Solutions for Chapter 4 Determinants Class 12 Maths Exercise 4.2

Class 12 Maths NCERT Solutions for Chapter 4 Determinants Exercise 4.2 

Determinants Exercise 4.2 Solutions

1. Using the property of determinants and without expanding,  prove that : 

Solution

Here, two columns of each determinant are identical.

2. Using the property of determinants and without expanding, prove that : 

Solution

3. Using the property of determinants and without expanding, prove that : 

Solution

4. Using the property of determinants and without expanding, prove that : 

Solution

5. Using the property of determinants and without expanding, prove that : 

Solution

Hence, the given result is proved. 

6. By using properties of determinants, show that: 

Solution 

We have, 

Here, the two rows R₁ and R₃ are identical.

∴ Δ = 0.

7. By using properties of determinants, show that : 

Solution

8. By using properties of determinants, show that : 

Solution

(i) Let Δ = 

Applying R₁ → R₁ → R₃ and R₂ → R₂ →R₃ , we have :

(ii) Let  Δ = 

Applying C₁ → C₁ → C₃ and C₂ → C₂ →C₃ , we have :

= (a - b)(b - c)(c - a)(a + b + c)
Hence, the given result is proved. 

9. By using properties of determinants, show that : 

= (x - y)(y - z)(z - x)(xy + yz + zx)

Solution

Let Δ = 
Applying R₂ → R₂ → R₁ and R₃ → R₃ → R₁ , we have :

= (x - y)(z - x)(z- y)[(-xz - yz) + (-x² – xy + x² )]
= -(x - y)(z -x)(z - y)(xy + yz + zx)
= (x - y)(y - z)(z - x)(xy + yz + zx)
Hence, the given result is proved.

10. By using properties of determinants, show that : 

Solution

(i)

(ii) 

11. By using properties of determinants, show that : 

Solution

12. By using properties of determinants show that : 

Solution

13. By using properties of determinants show that : 

Solution 

14. By using properties of determinants show that : 

Solution

15. Let A be a square matrix of order 3 × 3, then | kA| is equal to
(A) k|A|
(B) k² | A |
(C) k³ | A |
(D) 3k | A |

Solution

Let A = 
Then, 

The correct option is C. 

16. Which of the following is correct?
(A) Determinant is a square matrix.
(B) Determinant is a number associated to a matrix.
(C) Determinant is a number associated to a square matrix.
(D) None of the above.

Solution

We know that to every square matrix, A = [a_ij] of order n , we can associate a number called the determinant of square matrix A, where a_ij = (i, j)^th element of A. Thus, the determinant is a number associated to a square matrix. Hence, the correct option is C.