First slide

Introduction to Determinants

Question

If $f (α, β) = |\begin{array}{ccc} \cos α & - \sin α & 1 \\ \sin α & \cos α & 1 \\ \cos (α + β) & - \sin (α + β) & 1 \end{array}|$ , then

Moderate

A

f(300, 200) = f(400, 200)
B

f(200, 400) = f(200, 600)
C

f(100, 200) = f(200, 200)
D

none of these

Solution

$|\begin{array}{ccc} \cos α & - \sin α & 1 \\ \sin α & \cos α & 1 \\ \cos (α + β) & - \sin (α + β) & 1 \end{array}|$
$= |\begin{array}{ccc} \cos α & - \sin α & 1 \\ \sin α & \cos α & 1 \\ 0 & 0 & 1 + \sin β - \cos β \end{array}| [Applying R_{3} \to R_{3} - R_{1} (\cos β) + R_{2} (\sin β)]$
$= (1 + \sin β - \cos β) (\cos^{2} α + \sin^{2} α) = 1 + \sin β - \cos β, which is independent of α .$

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