Let α, β, γ be three real numbers and A=1cos⁡(β−α)cos⁡(γ−α)cos⁡(α−β)1cos⁡(γ−β)cos⁡(α−γ)cos⁡(β−γ)1 then

A
$A$ is singular
B
$A$ is non-singular
C
$A$ is orthogonal
D
none of these

Solution:

$| A | = |\begin{array}{ccc} 1 & \cos (β - α) & \cos (γ - α) \\ \cos (α - β) & 1 & \cos (γ - β) \\ \cos (α - γ) & \cos (β - γ) & 1 \end{array}|$
$= |\begin{array}{ccc} \cos α & \sin α & 0 \\ \cos β & \sin β & 0 \\ \cos γ & \cos γ & 0 \end{array}| |\begin{array}{ccc} \cos α & \sin α & 0 \\ \cos β & \sin β & 0 \\ \cos γ & \sin γ & 0 \end{array}| = 0$
$\Rightarrow A$ is singular.

Let $α, β, γ$ be three real numbers and $A = [\begin{array}{ccc} 1 & \cos (β - α) & \cos (γ - α) \\ \cos (α - β) & 1 & \cos (γ - β) \\ \cos (α - γ) & \cos (β - γ) & 1 \end{array}]$ then

Solution:

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