What does it mean to set the technology matrix of the Leontif Input Output Model to 0?

Question

I'm just beginning to learn about Leontif Input Output Systems and Linear Algebra. What is the significance of the solution if I set the technology matrix equal to the trivial solution (make the system of equations into a homogeneous linear system) $$ \left[ \begin{array}{ccc|c} \frac {-1}2 & \frac 13 & \frac 12 & 0 \\ \frac 14 & \frac{-2}3 & \frac14 & 0 \\ \frac 14 & \frac 13 & \frac{-3}4 & 0 \end{array} \right] $$ This is assuming that the original system is: $$\frac 12x + \frac 13y + \frac 13 z = x$$ $$\frac 14x + \frac 13y + \frac 14 z = y$$ $$\frac 14x + \frac 13y + \frac 14 z = z$$ Where $x, y, z$ are the production values of the three sectors of a simple economy. Amount supplied is equal to the amount demanded.