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Determinants Ex-4.2 Questions Answers

Subjects

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 C→ C3 + C2, wehave:

Here, two columns C1 and Care 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 R→ 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 Rare identical.

Δ = 0.

Question 7 :

By using properties ofdeterminants, show that:

Answer 7 :


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

Question 8 :

By using properties ofdeterminants, show that:

Answer 8 :


(i)   (ii) 


Solution

(i) 

Applying R1 → R1 − RandR2 → 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 − CandC2 → 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 − RandR3 → R3 − R1, we have:

Applying R3 → R3 + R2, we have:

Expanding along R3, we have: