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.
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 ...
Let be the production vector, form (A), we have
The total intermediate demand from all five sectors is given by
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
, 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 .
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 the unique solution of .
If we apply this theorem to our problem, we can conclude that exists and the production vector has nonnegative entries and is the unique solution of our system . Thus he answers you obtained in question 3 and question 4 should be same.