Linear algebra:Input output matrices>

So, for example, if the matrix is 3x3 and the p vector is <100, 200, 300>^T, then the output vector would be <10, 15, 20>^T and the input vector would be <90, 185, 280>^T. Does that make sense?In summary, the conversation discusses the terminology and concepts of input-output matrices in economics, specifically focusing on the terms output, gross production, net production, and consumption. The conversation also includes two sample questions and their solutions, which involve solving for input and output vectors using a production matrix. The output vector represents the gross production of industries, while the input vector represents the money fed into the consumption matrix
  • #1

Homework Statement

I am trying to make sense of some of the terminology.
Basically i get that each industry in an input matrix is dependant on each of those industries to some specified extent in order for production.

Is the output what each sector produces before we take into acount the consumption of each matrix? and is the production what we have left after consumption?

I will post two sample questions that i have been working on.

1)If the gross production for this economy is $10 million of tourism, $15 million of trans-
portation, and $20 million of services, what is the total value of the inputs consumed by
each sector during the production process?

Is the gross production what we have after consumption.Ie Cx=total consumption, and thus total production +consumption = total output?

2)If the total outputs of the tourism, transportation, and services sectors are $70 million, $50
million, and $60 million, respectively, what is the net production of each sector?

So in this part they have used the term output and net production,Now what is the output, is it everything produced "prior" to consumption of each sector? and what is the difference between net production and gross production.The book i have has like 1 page on input output, so i couild use some help, because i want to make sure i am interpreting everything correctly.

(i have not posted the consumption matrix because i am trying to get a better ubderstanding of the terminology and what it means, solving is pretty straightforward)

Homework Equations

The Attempt at a Solution

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  • #2
I'll take a stab at this, but you should also verify what I say against what your book says.

The input and output refer to vectors that are associated with your consumption matrix. An input vector represents the amounts of money fed into a consumption matrix in regard to some number of industries, and an output vector represents the money produced by each of these same industries. The output vector would represent (I believe) the gross production of these industries. The net production would be a vector of the gross production minus the cost to produce those amounts.

You haven't told us anything about the production matrix, so I'll make some assumptions. In the first problem, you mention 3 industry sectors, so I'll assume that the matrix is 3 x 3.

You're basically looking at the matrix equation Ax = y, where A is the matrix, x is the inputs, and y is the outputs.

For the first problem, you are supposed to solve for x in the equation Ax = <10, 15, 20>^T. Hopefully, you know something about matrices, so that should suggest something you can do to solve for x, the inputs.
  • #3
Hi, thanks for the reply.It did help me.

Can anyone help me verify this. I basically spent awhole day studying only this topic and i think i almost have it.

b)If the gross production for this economy is $10 million of tourism, $15 million of trans-portation, and $20 million of services, what is the total value of the inputs consumed by each sector during the production process?

So the total production will be what is produced prior to taking into acount "consumption". Thus A.p
where p is the production vector gives us the consumption "from each sectors production"
But the question is worded as what is consumed "by each sector' not what value of the production is consumed which would imply the total consumption from each three sectors from a given production output sector. So what i did in this case was represent the consumptions in matrix form and sum each column for the total amount of the "total productio: that each sector consumed.
  • #4
Without seeing the whole problem, I can't say for sure, but the p vector in the matrix product Ap is, I believe, the input vector, not the production vector. The total production would be the vector Ap.

1. What is a linear algebra input output matrix?

A linear algebra input output matrix is a mathematical representation of a system that takes in inputs and produces outputs. The matrix is used to describe the relationship between the inputs and outputs, and can be manipulated to solve for unknown variables.

2. How is a linear algebra input output matrix used in real life?

Linear algebra input output matrices are used in a variety of applications, including engineering, economics, computer science, and physics. They can be used to model and solve real-life problems such as predicting future stock prices, optimizing production processes, and designing computer algorithms.

3. What are the basic operations that can be performed on a linear algebra input output matrix?

The basic operations that can be performed on a linear algebra input output matrix include addition, subtraction, multiplication, and inversion. These operations can be used to manipulate the matrix and solve for unknown variables.

4. What is the difference between a square and non-square linear algebra input output matrix?

A square linear algebra input output matrix has an equal number of rows and columns, while a non-square matrix has different numbers of rows and columns. Square matrices have special properties that make them easier to work with, but non-square matrices can still be used to model and solve systems.

5. How is a linear algebra input output matrix related to other mathematical concepts?

Linear algebra input output matrices are closely related to other mathematical concepts, such as vectors, determinants, and systems of linear equations. Matrices can be used to represent and solve these concepts, making them an important tool in mathematics and other fields.

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