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.
Question
3
:
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.
Question
5
:
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.
Question
6
:
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:
Question
8
:
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: