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