# NCERT solution class 12 chapter 4 Determinants exercise 4.2 mathematics part 1

## EXERCISE 4.2

#### Question 1:

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

Using the property of determinants and without expanding, prove that:  Here, the two rows R1 and R3 are identical. Δ = 0.

#### Question 3:

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

Using the property of determinants and without expanding, prove that:  By applying C→ C3 + C2, we have: Here, two columns C1 and Care proportional. Δ = 0.

#### Question 5:

Using the property of determinants and without expanding, prove that:  Applying R2 → R2 − R3, we have: Applying R1 ↔R3 and R2 ↔R3, we have:  Applying R→ R1 − R3, we have: Applying R1 ↔R2 and R2 ↔R3, we have: From (1), (2), and (3), we have: Hence, the given result is proved.

#### Question 6:

By using properties of determinants, show that: We have,  Here, the two rows R1 and Rare identical.

∴Δ = 0.

#### Question 7:

By using properties of determinants, show that:  Applying R→ R2 + R1 and R→ R3 + R1, we have: #### Question 8:

By using properties of determinants, show that:

(i) (ii) (i) Applying R1 → R1 − Rand R2 → R2 − R3, we have: Applying R1 → R1 + R2, we have: Expanding along C1, we have: Hence, the given result is proved.

(ii) Let .

Applying C1 → C1 − Cand C2 → C2 − C3, we have: Applying C1 → C1 + C2, we have: Expanding along C1, we have: Hence, the given result is proved.

#### Question 9:

By using properties of determinants, show that:  Applying R2 → R2 − Rand R3 → R3 − R1, we have: Applying R3 → R3 + R2, we have: Expanding along R3, we have: Hence, the given result is proved.

#### Question 10:

By using properties of determinants, show that:

(i) (ii) (i) Applying R1 → R1 + R+ R3, we have: Applying C2 → C2 − C1, C3 → C3 − C1, we have: Expanding along C3, we have: Hence, the given result is proved.

(ii) Applying R1 → R1 + R+ R3, we have: Applying C2 → C2 − Cand C3 → C3 − C1, we have: Expanding along C3, we have: Hence, the given result is proved.

#### Question 11:

By using properties of determinants, show that:

(i) (ii) (i) Applying R1 → R1 + R+ R3, we have: Applying C2 → C2 − C1, C3 → C3 − C1, we have: Expanding along C3, we have: Hence, the given result is proved.

(ii) Applying C1 → C1 + C+ C3, we have: Applying R2 → R2 − Rand R3 → R3 − R1, we have: Expanding along R3, we have: Hence, the given result is proved.

#### Question 12:

By using properties of determinants, show that:  Applying R1 → R1 + R+ R3, we have: Applying C2 → C2 − Cand C3 → C3 − C1, we have: Expanding along R1, we have: Hence, the given result is proved.

#### Question 13:

By using properties of determinants, show that:  Applying R1 → R1 + bRand R2 → R2 − aR3, we have: Expanding along R1, we have: #### Question 14:

By using properties of determinants, show that:  Taking out common factors ab, and c from R1, R2, and Rrespectively, we have: Applying R2 → R2 − Rand R3 → R3 − R1, we have: Applying C1 → aC1, C→ bC2, and C3 → cC3, we have: Expanding along R3, we have: Hence, the given result is proved.

#### Question 15:

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

A. B. C. D. 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

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. 