The number of positive integral solutions of the equation $\begin{vmatrix} x^2+1 & x y & x z^2 \\ x y & y^2+1 & y z^2 \\ x z^2 & y z^2 & z^2+1 \end{vmatrix} = 11$ is are
Step-by-Step Solution
Key Concept: Special determinant structures can often be factored or simplified; recognizing patterns reduces computation.
The given determinant can be factored. Notice that the matrix has a special structure. Computing the determinant: $\begin{vmatrix} x^2+1 & xy & xz^2 \\ xy & y^2+1 & yz^2 \\ xz^2 & yz^2 & z^2+1 \end{vmatrix} = (1 + x^2 + y^2 + z^4)$. Setting this equal to 11 gives $1 + x^2 + y^2 + z^4 = 11$, so $x^2 + y^2 + z^4 = 10$. For positive integers, we need $x, y, z \geq 1$. Since $z^4$ grows rapidly, $z = 1$ gives $z^4 = 1$, so $x^2 + y^2 = 9$. The only solution is $(x,y) = (3,0)$ or permutations, but we need positive integers, so there are no solutions.
Correct Answer: 1