## NCERT Solutions for Class 12 Maths Chapter 4 – Determinants Ex 4.2

#### Page No 119:

#### Question 1:

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

#### Answer:

#### Question 2:

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

#### Answer:

Here, the two rows R

_{1}and R_{3}are identical.
Δ = 0.

#### Question 3:

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

#### Answer:

#### Question 4:

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

#### Answer:

By applying C

_{3 }→ C_{3}+ C_{2, }we have:
Here, two columns C

_{1}and C_{3 }are proportional.
Δ = 0.

#### Question 5:

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

#### Answer:

Applying R

_{2}→ R_{2}− R_{3}, we have:
Applying R

_{1}↔R_{3}and R_{2}↔R_{3}, we have:
Applying R

_{1 }→ R_{1}− R_{3}, we have:
Applying R

_{1}↔R_{2}and R_{2}↔R_{3}, we have:
From (1), (2), and (3), we have:

Hence, the given result is proved.

#### Page No 120:

#### Question 6:

By using properties of determinants, show that:

#### Answer:

We have,

Here, the two rows R

_{1}and R_{3 }are identical.
∴Δ = 0.

#### Question 7:

By using properties of determinants, show that:

#### Answer:

Applying R

_{2 }→ R_{2}+ R_{1}and R_{3 }→ R_{3}+ R_{1}, we have:#### Question 8:

By using properties of determinants, show that:

(i)

(ii)

#### Answer:

(i)

Applying R

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

_{1}→ R_{1}+ R_{2}, we have:
Expanding along C

_{1}, we have:
Hence, the given result is proved.

(ii) Let.

Applying C

_{1}→ C_{1}− C_{3 }and C_{2}→ C_{2}− C_{3}, we have:
Applying C

_{1}→ C_{1}+ C_{2}, we have:
Expanding along C

_{1}, we have:
Hence, the given result is proved.

#### Question 9:

By using properties of determinants, show that:

#### Answer:

Applying R

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

_{3}→ R_{3}+ R_{2}, we have:
Expanding along R

_{3}, we have:
Hence, the given result is proved.

#### Question 10:

By using properties of determinants, show that:

(i)

(ii)

#### Answer:

(i)

Applying R

_{1}→ R_{1}+ R_{2 }+ R_{3}, we have:
Applying C

_{2}→ C_{2}− C_{1}, C_{3}→ C_{3}− C_{1}, we have:
Expanding along C

_{3}, we have:
Hence, the given result is proved.

(ii)

Applying R

_{1}→ R_{1}+ R_{2 }+ R_{3}, we have:
Applying C

_{2}→ C_{2}− C_{1 }and C_{3}→ C_{3}− C_{1}, we have:
Expanding along C

_{3}, we have:
Hence, the given result is proved.

#### Question 11:

By using properties of determinants, show that:

(i)

(ii)

#### Answer:

(i)

Applying R

_{1}→ R_{1}+ R_{2 }+ R_{3}, we have:
Applying C

_{2}→ C_{2}− C_{1}, C_{3}→ C_{3}− C_{1}, we have:
Expanding along C

_{3}, we have:
Hence, the given result is proved.

(ii)

Applying C

_{1}→ C_{1}+ C_{2 }+ C_{3}, we have:
Applying R

_{2}→ R_{2}− R_{1 }and R_{3}→ R_{3}− R_{1}, we have:
Expanding along R

_{3}, we have:
Hence, the given result is proved.

#### Page No 121:

#### Question 12:

By using properties of determinants, show that:

#### Answer:

Applying R

_{1}→ R_{1}+ R_{2 }+ R_{3}, we have:
Applying C

_{2}→ C_{2}− C_{1 }and C_{3}→ C_{3}− C_{1}, we have:
Expanding along R

_{1}, we have:
Hence, the given result is proved.

#### Question 13:

By using properties of determinants, show that:

#### Answer:

Applying R

_{1}→ R_{1}+*b*R_{3 }and R_{2}→ R_{2}−*a*R_{3}, we have:
Expanding along R

_{1}, we have:#### Question 14:

By using properties of determinants, show that:

#### Answer:

Taking out common factors

*a*,*b*, and*c*from R_{1}, R_{2}, and R_{3 }respectively, we have:
Applying R

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

_{1}→*a*C_{1}, C_{2 }→*b*C_{2, }and C_{3}→*c*C_{3}, we have:
Expanding along R

_{3}, we have:
Hence, the given result is proved.

#### Question 15:

Choose the correct answer.

Let

*A*be a square matrix of order 3 × 3, then is equal to**A.**

**B.**

**C.**

**D.**

#### Answer:

**Answer: C**

*A*is a square matrix of order 3 × 3.

Hence, the correct answer is C.

#### Question 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 these

#### Answer:

**Answer: C**

We know that to every square matrix, of order

*n*. We can associate a number called the determinant of square matrix*A*, where element of*A*.
Thus, the determinant is a number associated to a square matrix.

Hence, the correct answer is C.

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