## EXERCISE 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.