Question

The number of ordered triplets of positive integral solutions of $\left|\begin{array}{ccc}{y}^3+1& y^{2}z& y^{2}x\\ y{z}^2& z^3+1& z^{2}x\\ y{x}^2& x^{2}z& x^3+1\end{array}\right|=11$ is

Answer 1

Let ,

$A=\left|\begin{array}{ccc}{y}^3+1& y^{2}z& y^{2}x\\ y{z}^2& z^3+1& z^{2}x\\ y{x}^2& x^{2}z& x^3+1\end{array}\right|$

Expanding the matrix by first row, we get,
$A=(y^3+1)[z^3{x}^3+z^3+1+x^3-z^3{x}^3]-y^{2}z[y{x}^3{z}^2+y{z}^2-y{x}^3{z}^2]+y^{2}x[y{x}^2{z}^3-y{x}^2{z}^3-y{x}^2]$

$=x^3+y^3+z^3+1=11$

$x^3+y^3+z^3=10$

We can have ordered pairs of $(2,1,1)$ i.e.
$x=2,y=1,z=1,x=1,y=1,z=2andx=1,y=2,z=1$

Hence, there are three ordered triplets of positive integral solutions.