**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 R

_{1}and R

_{3}are identical.

∴ Î” = 0.

**7. By using properties of determinants, show that :**

**Solution**

**8. By using properties of determinants, show that :**

**Solution**

(i) Let Î” =

Applying R_{1} → R_{1} → R_{3} and R_{2} → R_{2} →R_{3} , we have :

(ii) Let Î” =

Applying C_{1} → C_{1} → C_{3} and C_{2} → C_{2} →C_{3} , 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)

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

**Solution**

Let Î” =

Applying R_{2} → R_{2} → R_{1} and R_{3} → R_{3} → R_{1} , we have :

= (x - y)(z - x)(z- y)[(-xz - yz) + (-x^{2} – xy + x^{2} )]

= -(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) k

(C) k

(D) 3k | A |

(A) k|A|

(B) k

^{2}| A |(C) k

^{3}| A |(D) 3k | A |

**Solution**

Let A =

Then,

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.

(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.