Determinants
Miscellaneous Solutions
Question 1:
Prove that the determinant is independent of θ.
Answer
Hence, Δ is independent of θ.
Question 2: Without expanding the determinant, prove that
Answer
Hence, the given result is proved.
Question 3: Evaluate
Answer
Expanding along C_{3}, we have:
Question 4: If a, b and c are real numbers, and
,
Show that either a + b + c = 0 or a = b = c.
Answer
Expanding along R_{1}, we have:
Hence, if Δ = 0, then either a + b + c = 0 or a = b = c.
Question 5: Solve the equations
Answer
Question 6: Prove that
Answer
Expanding along R3, we have:
Hence, the given result is proved.
Question 8: Let verify that
Question 9: Evaluate
Answer
Expanding along R_{1}, we have:
Question 10: Evaluate
Answer
Question 11: Using properties of determinants, prove that:
Answer
Hence, the given result is proved.
Question 12: Using properties of determinants, prove that:
Answer
Expanding along R_{3}, we have:
Hence, the given result is proved.
Question 13: Using properties of determinants, prove that:
Answer
Hence, the given result is proved.
Question 14: Using properties of determinants, prove that:
Answer
Hence, the given result is proved.
Question 15: Using properties of determinants, prove that:
Answer
Hence, the given result is proved.
Question 16: Solve the system of the following equations
Answer
This system can be written in the form of AX = B, where
Question 17: Choose the correct answer. If a, b, c, are in A.P., then the determinant
A. 0
B. 1
C. x
D. 2x
Answer
Answer: A
Here, all the elements of the first row (R1) are zero.
Hence, we have Δ = 0.
The correct answer is A.
Question 18: Choose the correct answer. If x, y, z are nonzero real numbers, then the inverse of matrix
is
A.
B.
C.
D.
Answer: A
The correct answer is A.
Question 19: Choose the correct answer.
Let , where 0 ≤ θ≤ 2π, then
A). Det (A) = 0
B). Det (A) (2, ∞)
C). Det (A) (2, 4)
D). Det (A) [2, 4]
Answer: D
The correct answer is D.
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