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(D) Let $\Delta = \left| {\begin{array}{*{20}{c}}{y + z}&{x - z}&{x - y}\{y - z}&{z + x}&{y - x}\{z - y}&{z - x}&{x + y}\end{array}} ight|$.Apply

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$8$

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(D) Let $\Delta = \left| {\begin{array}{*{20}{c}}{y + z}&{x - z}&{x - y}\{y - z}&{z + x}&{y - x}\{z - y}&{z - x}&{x + y}\end{array}} ight|$.Apply row operations ${R_2} 	o {R_2} + {R_1}$ and ${R_3} 	o {R_3} + {R_1}$:$\Delta = \left| {\begin{array}{*{20}{c}}{y + z}&{x - z}&{x - y}\{2y}&{2x}&0\{2z}&0&{2x}\end{array}} ight|$.Taking $2$ common from $R_2$ and $R_3$:$\Delta = 4 \left| {\begin{array}{*{20}{c}}{y + z}&{x - z}&{x - y}\y&x&0\z&0&x\end{array}} ight|$.Expanding along the first row:$\Delta = 4[(y + z)(x^2 - 0) - (x - z)(xy - 0) + (x - y)(0 - zx)]$$\Delta = 4[x^2y + zx^2 - (x^2y - xyz) - (xzx - xyz)]$$\Delta = 4[x^2y + zx^2 - x^2y + xyz - x^2z + xyz]$$\Delta = 4[2xyz] = 8xyz$.Comparing with $kxyz$,we get $k = 8$.

If $\left| {\begin{array}{*{20}{c}}{y + z}&{x - z}&{x - y}\\{y - z}&{z + x}&{y - x}\\{z - y}&{z - x}&{x + y}\end{array}} \right| = kxyz$,then the value of $k$ is

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