First slide

Applications of determinant

Question

Let $A (θ) = (\begin{array}{cc} \sin θ & i \cos θ \\ i \cos θ & \sin θ \end{array}),$ then

Moderate

A

$A (θ)^{- 1} = A (- θ)$
B

$A (θ)^{- 1} = A (π - θ)$
C

$A (θ)^{- 1}$ does not exist
D

$A (θ)^{2} = A (2 θ)$

Solution

$\begin{array}{l} | A (θ) | = \sin^{2} θ - i^{2} \cos^{2} θ = 1 \\ ∴ A (θ)^{- 1} = (\begin{array}{cc} \sin θ & - i \cos θ \\ - i \cos θ & \sin θ \end{array}) \\ = A (π - θ) \end{array}$

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