What would the determinant of the input-output matrix represent? What is the intuition? I know that the determinant of a 2x2 matrix represents the oriented area of the parallelogram represented by the vectors in the matrix. For 3x3 or higher order matrix it represents the oriented volume. I am trying to get such an intuitive understanding of the determinant of the input-output matrix used in macroeconomics. What could it represent? Does it represent the size of the economy? We usually measure the size of the economy by measuring it's GDP. If it represents the size of the economy, could this determinant be a better measure of the size of the economy than GDP?
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.