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(C) Let the given determinant be $D = \left| \begin{matrix} 0 & x - y & x - z \ y - x & 0 & y - z \ z - x & z - y & 0 \end{matrix} ight|$.Observe

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$0$

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(C) Let the given determinant be $D = \left| \begin{matrix} 0 & x - y & x - z \ y - x & 0 & y - z \ z - x & z - y & 0 \end{matrix} ight|$.Observe that the determinant is of the form $A = [a_{ij}]$ where $a_{ij} = -a_{ji}$ for all $i, j$. Specifically,$a_{11} = 0, a_{22} = 0, a_{33} = 0$,and $a_{12} = -(a_{21})$,$a_{13} = -(a_{31})$,$a_{23} = -(a_{32})$.This is the definition of a skew-symmetric matrix.The determinant of a skew-symmetric matrix of odd order ($n 	imes n$ where $n$ is odd) is always $0$.Since this is a $3 	imes 3$ matrix (order $3$,which is odd),the value of the determinant is $0$.

The value of the determinant $\left| \begin{matrix} 0 & x - y & x - z \\ y - x & 0 & y - z \\ z - x & z - y & 0 \end{matrix} \right|$ is:

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