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(B) The system of equations is homogeneous,represented by $AX = 0$,where $A$ is the coefficient matrix.The determinant of the coefficient matrix $A$ i

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infinitely many solutions

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(B) The system of equations is homogeneous,represented by $AX = 0$,where $A$ is the coefficient matrix.The determinant of the coefficient matrix $A$ is given by:$|A| = \begin{vmatrix} 1 & \cos \gamma & \cos \beta \ \cos \gamma & 1 & \cos \alpha \ \cos \beta & \cos \alpha & 1 \end{vmatrix}$Expanding the determinant:$|A| = 1(1 - \cos^2 \alpha) - \cos \gamma(\cos \gamma - \cos \alpha \cos \beta) + \cos \beta(\cos \gamma \cos \alpha - \cos \beta)$$|A| = 1 - \cos^2 \alpha - \cos^2 \gamma + \cos \alpha \cos \beta \cos \gamma + \cos \alpha \cos \beta \cos \gamma - \cos^2 \beta$$|A| = 1 - \cos^2 \alpha - \cos^2 \beta - \cos^2 \gamma + 2 \cos \alpha \cos \beta \cos \gamma$Given $\alpha + \beta + \gamma = 2\pi$,we have $\gamma = 2\pi - (\alpha + \beta)$,so $\cos \gamma = \cos(\alpha + \beta)$ and $\sin \gamma = -\sin(\alpha + \beta)$.Using the identity for the determinant of this specific symmetric matrix,it is known that $|A| = 1 - \cos^2 \alpha - \cos^2 \beta - \cos^2 \gamma + 2 \cos \alpha \cos \beta \cos \gamma = 0$ when $\alpha + \beta + \gamma = 2n\pi$.Since $|A| = 0$,the system of homogeneous equations has infinitely many solutions.

If $\alpha+\beta+\gamma=2 \pi$,then the system of equations
$x+(\cos \gamma) y+(\cos \beta) z=0$
$(\cos \gamma) x+y+(\cos \alpha) z=0$
$(\cos \beta) x+(\cos \alpha) y+z=0$
has :

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If $\alpha+\beta+\gamma=2 \pi$,then the system of equations$x+(\cos \gamma) y+(\cos \beta) z=0$$(\cos \gamma) x+y+(\cos \alpha) z=0$$(\cos \beta) x+(\cos \alpha) y+z=0$has :