# Given the technology matrix, A, and the demand vector, D, below: A = [ .1   .3 ]                          D = [ 15 ]       [ 2     0 ]                            ...

Given the technology matrix, A, and the demand vector, D, below:

A = [ .1   .3 ]                          D = [ 15 ]

[ 2     0 ]                                 [ 3  ]

Compute the production vector, X.

sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

You should evaluate the production vector `X` , using the following formula that relates the technology matrix and and demand vector, such that:

`X = (I-A)^(-1)*D => (I-A)*X = D`

You need to evaluate I-A such that:

`I-A = ((1,0),(0,1)) - ((0.1,0.3),(2,0))`

`I-A = ((1-0.1,-0.3),(-2,1))`

You need to perform the multiplication `(I-A)*X = D` , hence, `X` is a column vector `X = ((a),(b))` , such that:

`((0.9,-0.3),(-2,1))((a),(b)) = ((15),(3))` => `{(0.9a - 0.3b = 15),(-2a + b = 3):}`

Replacing `3 + 2a` for b in the top equation yields:

`0.9a - 0.3(3 + 2a) = 15 => 0.9a - 0.9 - 0.6a = 15`

`0.3a - 0.9 = 15 => 0.3a = 15+0.9 => 0.3a = 15.9 => a = 53`

`b = 3 + 2a => b = 3 + 106 => b = 109`

Hence, evaluating the production vector X, yields `X = ((53),(109)).`