Page 1
Exercise 4.2
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
Page 2
Exercise 4.2
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
XII
Here, the two rows R1 and R3 are identical.
? = 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
Page 3
Exercise 4.2
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
XII
Here, the two rows R1 and R3 are identical.
? = 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 + C2, we have:
Here, two columns C1 and C3 are proportional.
? = 0.
Question 5:
Using the property of determinants and without expanding, prove that:
Answer
Applying R2 ? R2 - R3, we have:
Page 4
Exercise 4.2
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
XII
Here, the two rows R1 and R3 are identical.
? = 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 + C2, we have:
Here, two columns C1 and C3 are proportional.
? = 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:
Page 5
Exercise 4.2
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
XII
Here, the two rows R1 and R3 are identical.
? = 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 + C2, we have:
Here, two columns C1 and C3 are proportional.
? = 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.
Question 6:
By using properties of determinants, show that:
Answer
We have,
Here, the two rows R1 and R3 are identical.
? = 0.
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