What would the determinant of the input-output matrix represent? What is the intuition?
Your conjecture seems unlikely. For example the unit matrix would show an economy where no new goods can be produced (it takes 1 unit of something to make 1 unit of that same thing). Yet the determinant of this matrix is 1. If we were to multiply the $3 \times 3$ unit matrix by say 10, the new determinant would be a 1000. But the economy did not get better, if anything it got worse.