First slide

Introduction to Determinants

Question

If $θ \in R$ , then maximum value of $Δ = |\begin{array}{ccc} 1 & 1 & 1 \\ 1 & 1 + \sin θ & 1 \\ 1 & 1 & 1 + \cos θ \end{array}|$ is

Moderate

A

$\sqrt{3} / 2$
B

1/2
C

$1 / \sqrt{2}$
D

None of these

Solution

Applying $C_{2} \to C_{2} - C_{1} and C_{3} \to C_{3} - C_{1}$ , we get
$Δ = |\begin{array}{ccc} 1 & 0 & 0 \\ 1 & \sin θ & 0 \\ 1 & 0 & \cos θ \end{array}| = \sin θcos θ = (\sin 2 θ) / 2$
So, maximum value of $Δ$ is 1/2.

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