Leontief Economic Model - Input Output analysis: Mfg, Electricity, and Transport

[Njords]

Company operates in three divisions: Manufacturing, Electricity, Transport.
The divisions are referred to by M, E, and T respectively.
M requires 0.4 from M, 0.3 from P, and 0.1 from L to produce 1.0;
P requires 0.8 from M;
L requires 0.8 from M and 1.4 from P.

(a) Write down the input-output matrix (A).
(b) Are all the divisions protable?

(c) Is matrix A a productive matrix?

(d) The external demand in a particular week is d = [5000 2000 1000]T .
What output will supply demand?

(e) The external demand in another week is d = [0 0 1000]T . What output
will supply demand and what is the profit?

________________________________________________________________________________________________________________________________________

a)
M E T

0.4	0.8	0.8
0.3	0	1.4
0.1	0	0

b)

0.4	0.8	0.8
0.3	0	1.4
0.1	0	0

0.8 0.8 2.2 Implies Division T not profitable since input of 2.2 to get 1.0 output. ( sound good?)

c) I think here to show matrix A is productive I must demonstrate a reasonable production to satisfy demand matrix d.
in other words (I-A)^-1 must not contain any negative values.

. . . . .\(\displaystyle \left(\begin{array}{ccc}0.6&-0.8&-0.8\\-0.3&1&-1.4\\-0.1&0&1\end{array}\right)^{-1}\, =\, \left(\begin{array}{ccc}5.95238...&4.76190...&11.42857...\\2.61905...&3.09524...&6.42857...\\0.59524...&0.47619...&2.14286...\end{array}\right)\)

therefore A is productive............. hopefully so far so good?

d)demand is given by transpose of vector [5000,2000,1000] which is a column 3x1

and out put will be x = (I-A)^-1 x d or..

. . . . .\(\displaystyle \left(\begin{array}{ccc}5.95238&4.76190&11.42857\\2.61905&3.09524&6.42857\\0.59524&0.47619&2.14286 \end{array}\right)\, \left( \begin{array}{ccc}5000\\2000\\1000\end{array} \right)\, =\, \left(\begin{array}{ccc}50714.27\\25714.3\\6071.44\end{array}\right)\)

(I-A)^-1 d = x

e) similar as d) above this time the demand vector is transpose of [0,0,1000]

. . . . .\(\displaystyle \left(\begin{array}{ccc}5.95238&4.76190&11.42857\\2.61905&3.09524&6.42857\\0.59524&0.47619&2.14286 \end{array}\right)\, \left( \begin{array}{ccc}0\\0\\1000\end{array} \right)\, =\, \left(\begin{array}{ccc}11428.57\\6428.57\\2142.86\end{array}\right)\)

.......... and what is the profit? This is where I'm stuck and not sure where to go, assuming all else I have done above is right.