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

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 R₁ and R₃ 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 C₃ → C₃ + C_2, we have:

Here, two columns C₁ and C₃ are proportional.

Δ = 0.

Question 5:

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

Answer:

Applying R₂ → R₂ − R₃, we have:

Applying R₁ ↔R₃ and R₂ ↔R₃, we have:

Applying R₁ → 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.

Page No 120:

Question 6:

By using properties of determinants, show that:

Answer:

We have,

Here, the two rows R₁ and R₃ are identical.

∴Δ = 0.

Question 7:

By using properties of determinants, show that:

Answer:

Applying R₂ → R₂ + R₁ and R₃ → R₃ + R₁, we have:

Question 8:

By using properties of determinants, show that:

(i) 

(ii) 

Answer:

(i) 

Applying R₁ → R₁ − R₃ and 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 − C₃ and 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:

Answer:

Applying R₂ → R₂ − R₁ and 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) 

Answer:

(i) 

Applying R₁ → R₁ + R₂ + 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 + R₂ + R₃, we have:

Applying C₂ → C₂ − C₁ and 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) 

Answer:

(i) 

Applying R₁ → R₁ + R₂ + 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 + C₂ + C₃, we have:

Applying R₂ → R₂ − R₁ and R₃ → R₃ − R₁, we have:

Expanding along R₃, we have:

Hence, the given result is proved.

Page No 121:

Question 12:

By using properties of determinants, show that:

Answer:

Applying R₁ → R₁ + R₂ + R₃, we have:

Applying C₂ → C₂ − C₁ and 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:

Answer:

Applying R₁ → R₁ + bR₃ and R₂ → R₂ − aR₃, we have:

Expanding along R₁, we have:

Question 14:

By using properties of determinants, show that:

Answer:

Taking out common factors a, b, and c from R₁, R₂, and R₃ respectively, we have:

Applying R₂ → R₂ − R₁ and R₃ → R₃ − R₁, we have:

Applying C₁ → aC₁, C₂ → bC_2, and C₃ → cC₃, we have:

Expanding along R₃, 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.