Find the cofactors of elements $a_{12}, a_{22}, a_{32}$ , respectively of the matrix $[\begin{array}{ccc} 1 & \sin θ & 1 \\ - \sin θ & 1 & \sin θ \\ - 1 & - \sin θ & 1 \end{array}]$

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$\begin{aligned} 0, 2, - 2 \sin θ \end{aligned}$

$\begin{aligned} 2, 0, 2 \sin θ \end{aligned}$

$2, 0, - 2 \sin θ$

$- 2 \sin θ, 2, 0$

answer is A.

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Detailed Solution

$\begin{array}{l} M_{12} = |\begin{array}{cc} - \sin θ & \sin θ \\ - 1 & 1 \end{array}| = - \sin θ + \sin θ = 0 \\ \Rightarrow c_{12} = - M_{12} = 0 \end{array}$

$\begin{array}{l} M_{22} = |\begin{array}{cc} 1 & 1 \\ - 1 & 1 \end{array}| = 1 + 1 = 2 \Rightarrow c_{22} = M_{22} = 2 \\ M_{32} = |\begin{array}{cc} 1 & 1 \\ - \sin θ & \sin θ \end{array}| = \sin θ + \sin θ = 2 \sin θ \\ c_{32} = - M_{32} = - 2 \sin θ \end{array}$