<p>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.</p>

CC BY-SA 3.0
2016-08-03T22:31:26.417Z
2016-08-04T02:41:23.170Z
2016-08-04T02:41:23.170Z
Giskard
<p>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? </p>

CC BY-SA 3.0
2016-08-03T23:03:47.710Z
If the input-output matrix was $A$ then you might do better looking at the determinant of $I-A$ than $A$ itself
Henry
CC BY-SA 3.0
2016-08-04T01:46:27.883Z
? https://books.google.com.hk/books?id=viHaAgAAQBAJ&pg=PA61&lpg=PA61&dq=determinant+input-output+matrix&source=bl&ots=gpHhezjV6U&sig=xd_05TkfipK4EYoGjAijkSP7sYw&hl=en&sa=X&ved=0ahUKEwii24fJ0abOAhVGlJQKHcnQBfEQ6AEISTAI#v=onepage&q=determinant%20input-output%20matrix&f=false
BCLC
CC BY-SA 3.0
2016-08-04T01:46:44.760Z
? http://www.sjsu.edu/faculty/watkins/inputoutput.htm
BCLC
<macroeconomics><mathematical-economics><gdp><input-output>
What would the determinant of the input-output matrix represent? What is the intuition?
user9028
# What would the determinant of the input-output matrix represent? What is the intuition?

(Asked by: username_0 on 2016-08-03)

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?

**Tags:** macroeconomics, mathematical-economics, gdp, input-output
> **Comments:**
> **username_1**: If the input-output matrix was $A$ then you might do better looking at the determinant of $I-A$ than $A$ itself  
> **username_2**: ? https://books.google.com.hk/books?id=viHaAgAAQBAJ&pg=PA61&lpg=PA61&dq=determinant+input-output+matrix&source=bl&ots=gpHhezjV6U&sig=xd_05TkfipK4EYoGjAijkSP7sYw&hl=en&sa=X&ved=0ahUKEwii24fJ0abOAhVGlJQKHcnQBfEQ6AEISTAI#v=onepage&q=determinant%20input-output%20matrix&f=false  
> **username_2**: ? http://www.sjsu.edu/faculty/watkins/inputoutput.htm  

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## 1 answers

### Answer by username_3 on 2016-08-03. Score: 1

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.


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