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The number of positive integral solutions of the equation $\left| {\begin{array}{*{20}{c}}{{x^3} + 1}&{{x^2}y}&{{x^2}z}\\{x{y^2}}&{{y^3} + 1}&{{y^2}z}\\{x{z^2}}&{y{z^2}}&{{z^3} + 1}\end{array}} \right|$ $= 11$ is
$0$
$3$
$6$
$12$
Solution
$\left| {\,\begin{array}{*{20}{c}}{{x^4} + x}&{{x^3}y}&{{x^3}z}\\{x{y^3}}&{{y^4} +y}&{{y^3}z}\\{x{z^3}}&{y{z^3}}&{{z^4} + z}\end{array}\,} \right|$ $= 11$
$\frac{1}{{xyz}}\left| {\,\begin{array}{*{20}{c}}{{x^3} + 1}&{{x^3}}&{{x^3}}\\ {{y^3}}&{{y^3} + 1}&{{y^3}}\\{{z^3}}&{{z^3}}&{{z^3} + 1}\end{array}\,} \right|$ $= 11$
use $R_1\rightarrow R_1 + R_2 + R_3$
$D =$ $(x^3 + y^3 + z^3 + 1)$ $\left| {\,\begin{array}{*{20}{c}}1&1&1\\{{y^3}}&{{y^3} + 1}&{{y^3}}\\{{z^3}}&{{z^3}}&{{z^3} + 1}\end{array}\,} \right|$ $= 11$
hence $x^3 + y^3 + z^3 = 10$
$(2, 1, 1) , (1, 2, 1) , (1, 1, 2)$