Let A = begin{bmatrix} -cot theta & operatorn | vedclass

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(B) Given $A = \begin{bmatrix} -\cot 	heta & \operatorname{cosec} 	heta \ \operatorname{cosec} 	heta & -\cot 	heta \end{bmatrix}$.First,calculate

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$	heta_1 = \frac{\pi}{2}$,such $	heta_2$ does not exist

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(B) Given $A = \begin{bmatrix} -\cot 	heta & \operatorname{cosec} 	heta \ \operatorname{cosec} 	heta & -\cot 	heta \end{bmatrix}$.First,calculate the determinant $|A| = (-\cot 	heta)(-\cot 	heta) - (\operatorname{cosec} 	heta)(\operatorname{cosec} 	heta) = \cot^2 	heta - \operatorname{cosec}^2 	heta = -1$.Now,find $A^{-1} = \frac{1}{|A|} \operatorname{adj}(A) = -1 \begin{bmatrix} -\cot 	heta & -\operatorname{cosec} 	heta \ -\operatorname{cosec} 	heta & -\cot 	heta \end{bmatrix} = \begin{bmatrix} \cot 	heta & \operatorname{cosec} 	heta \ \operatorname{cosec} 	heta & \cot 	heta \end{bmatrix}$.For $A^{-1} = A$:$\begin{bmatrix} \cot 	heta & \operatorname{cosec} 	heta \ \operatorname{cosec} 	heta & \cot 	heta \end{bmatrix} = \begin{bmatrix} -\cot 	heta & \operatorname{cosec} 	heta \ \operatorname{cosec} 	heta & -\cot 	heta \end{bmatrix}$.Comparing elements,we get $\cot 	heta = -\cot 	heta$,which implies $2 \cot 	heta = 0$,so $\cot 	heta = 0$. This occurs at $	heta_1 = \frac{\pi}{2}$.For $A^{-1} + A = O$:$\begin{bmatrix} \cot 	heta & \operatorname{cosec} 	heta \ \operatorname{cosec} 	heta & \cot 	heta \end{bmatrix} + \begin{bmatrix} -\cot 	heta & \operatorname{cosec} 	heta \ \operatorname{cosec} 	heta & -\cot 	heta \end{bmatrix} = \begin{bmatrix} 0 & 2 \operatorname{cosec} 	heta \ 2 \operatorname{cosec} 	heta & 0 \end{bmatrix} = \begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix}$.This implies $2 \operatorname{cosec} 	heta = 0$,which means $\operatorname{cosec} 	heta = 0$. Since $\operatorname{cosec} 	heta = \frac{1}{\sin 	heta}$,it can never be zero for any real $	heta$. Thus,such $	heta_2$ does not exist.

Let $A = \begin{bmatrix} -\cot \theta & \operatorname{cosec} \theta \\ \operatorname{cosec} \theta & -\cot \theta \end{bmatrix}$. If $A^{-1} = A$ at $\theta = \theta_1$ and $A^{-1} + A = O$ at $\theta = \theta_2$,then which one of the following is true?

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