Chapter 4 Determinants Ex 4.2 Solutions
Question - 1 : - Using the property ofdeterminants and without expanding, prove that:
Answer - 1 : - 
Solution
Question - 2 : - Using the property ofdeterminants and without expanding, prove that-
Answer - 2 : - 
Solution
Here, the two rows R1 and R3 are identical.
Δ = 0.
Using the property ofdeterminants and without expanding, prove that:
Answer - 3 : -
Question - 4 : - Using the property ofdeterminants and without expanding, prove that:
Answer - 4 : -
By applying C3 → C3 + C2, wehave:
Here, two columns C1 and C3 are proportional.
Δ = 0.
Using the property ofdeterminants and without expanding, prove that:
Answer - 5 : -
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 resultis proved.
By using properties ofdeterminants, show that:
Answer - 6 : -
We have,
Here, the two rows R1 and R3 are identical.
∴Δ = 0.
Question - 7 : - By using properties ofdeterminants, show that:
Answer - 7 : -
Applying R2 → R2 + R1 and R3 → R3 + R1, we have:
By using properties ofdeterminants, show that:
Answer - 8 : -
(i) 
(ii) 
Solution
(i) 
Applying R1 → R1 − R3 andR2 → R2 − R3, we have:
Applying R1 → R1 + R2, we have:
Expanding along C1, we have:
Hence, the given resultis proved.
(ii) Let
Applying C1 → C1 − C3 andC2 → C2 − C3, we have:
Expanding along C1, we have:
Hence, the given resultis proved.
Question - 9 : - By using properties ofdeterminants, show that:
Answer - 9 : -
Applying R2 → R2 − R1 andR3 → R3 − R1, we have:
Applying R3 → R3 + R2, we have:
Expanding along R3, we have:
Hence, the given resultis proved.
By using properties ofdeterminants, show that:
Answer - 10 : -
(i) 
(ii) 
Solution
(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 resultis proved.
(ii) 
Applying R1 → R1 + R2 + R3, we have:
Applying C2 → C2 − C1 andC3 → C3 − C1, we have:
Expanding along C3, we have:
Hence, the given resultis proved.