Show that a consumption matrix as considered in Prob. 13 must have column sums 1 and always has the eigenvalue 1.

Prob 13

Suppose that three industries are interrelated so that their outputs are used as inputs by themselves, according to the 3 × 3 consumption matrix

where α_jk is the fraction of the output of industry k consumed (purchased) by industry j. Let P_j be the price charged by industry j for its total output. A problem is to find prices so that for each industry, total expenditures equal total income. Show that this leads to Ap = p, where p = [p₁   p₂    p₃]^T, and find a solution p with non negative p₁, p₂, p₃.

0.1 0.1 0.5 A = [ajk] =| 0.8 0.4 0.5 0.6 0.1