If a- b- c and d are complex numbers- then the determinant2 - a-b-c-d - a b-c da-b-c-d - 2a-bc-d - a bc-d-c da-ba b-c d - a bc-d-c da-b - 2 a b c dlend vmatrix isA- None of theseB- Dependent on a- b- c and dC- Independent of a- b- c and dD- Dependent on a- c and independent of b- d

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Evaluation of a Determinant

If a, b, c an...

Question

If a, b, c and d are complex numbers, then the determinant
$Δ = ∣ ∣ ∣ ∣ ∣ \begin{matrix} 2 & a + b + c + d & a b + c d a + b + c + d & 2 (a + b) (c + d) & a b (c + d) + c d (a + b) a b + c d & a b (c + d) + c d (a + b) & 2 a b c d \end{matrix} ∣ ∣ ∣ ∣ ∣$ is

Dependent on a, b, c and d

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Independent of a, b, c and d

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Dependent on a, c and independent of b, d

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None of these

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Solution

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Δ = ∣ ∣ ∣ ∣ ∣ \begin{matrix} 2 & a + b + c + d & a b + c d a + b + c + d & 2 (a + b) (c + d) & a b (c + d) + c d (a + b) a b + c d & a b (c + d) + c d (a + b) & 2 a b c d \end{matrix} ∣ ∣ ∣ ∣ ∣

∣ ∣ ∣ ∣ ∣ \begin{matrix} 2 & a + b + c + d & a b + c d a + b + c + d & 2 (a + b) (c + d) & a b (c + d) + c d (a + b) a b + c d & a b (c + d) + c d (a + b) & 2 a b c d \end{matrix} ∣ ∣ ∣ ∣ ∣

a, b, c, d

a + b + c + d = 2

\frac{1}{a b c} + \frac{1}{b c d} + \frac{1}{c d a} + \frac{1}{d a b}

∣ ∣ ∣ ∣ ∣ \begin{matrix} b + c - a - d & b c - a d & b c (a + d) - a d (b + c) c + a - b - d & c a - b d & c a (b + d) - b d (c + a) a + b - c - d & a b - c d & a b (c + d) - c d (a + b) \end{matrix} ∣ ∣ ∣ ∣ ∣ = - 2 (b - c) (c - a) (a - b) (a - d) (b - d) (c - d)

A, B, C, D

A^{T} = B C D;

B^{T} = C D A;

C^{T} = D A B

D^{T} = A B C .

S = A B C D,

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