This is the classic "Leontief Input-Output Model" problem. Please see your textbook or search on line to find more details about this economic model and how Linear Algebra played an important role in solving it. I am just going to give you the basic setup for your problem, you might need to fill in some details.

From the given, I can construct the following table that illustrates the inputs consumed per unit of output, please see the attachment.

Realize that the first column in above table is the consumption vector of Agriculture, to simplify the notation, let's denote it as (i.e. ); similarly, we can denote the consumption vectors of Manufacturing, Service, Entertainment and Mining as and respectively. And they are the 2nd, 3rd, 4th and 5th columns of above table respectively.

The intermediate demands of an economic sector can be found through multiplying the number of units by its corresponding consumption vector. For instance, in order to find the intermediate demand of Agriculture if it plans to produce 240 units, we calculate

Similarly, the intermediate demand of manufacturing 160 units is:

the intermediate demand of services 125 units is:

I think you get the idea and can finish the last two ...

Here the consumption matrix is just , namely,

Let be the production vector, form (A), we have

The total intermediate demand from all five sectors is given by

In order to answer this part, we need to know the Leontief Input-Output Model (Called Production Equation), which states that

Amt. Produced = Intermediate Demand + Final Demand

Mathematically, we have:

where is the production amount (level) vector, is the consumption matrix and is the final demand vector. Back to our problem, we have found out in part 2(B) and given

In order to determine the production levels needed, we just need to solve the following system for

Or equivalently,

, where is a 5x5 identity matrix. Note here we know the matrix and the vector , then we have a system of equations to solve for .

Realize that the system we obtained in (3) can also be solved by finding out the inverse matrix to (i.e. ) provided the inverse exists. If the inverse exists, we can determine the production level vector by

How do we know the matrix has an inverse or not? We have a nice theorem to use here. The theorem says: If and have nonnegative entries and it each column sum of is less than 1, then exists and the production vector has nonnegative entries and is theuniquesolution of .

If we apply this theorem to our problem, we can conclude that exists and the production vector has nonnegative entries and is theuniquesolution of our system . Thus he answers you obtained in question 3 and question 4 should be same.

It is your turn to finish it...

Roy