First slide

Applications of determinant

Question

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

Moderate

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.

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