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.

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 + C2, we have:

Here, two columns C1 and C3 are proportional.

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 → bC2, 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.