(4) Three firms, F₁, F2, F3 share the market for a certain commodity. F₁ has 20%, F₂ has 60% and F3 has 20%. In the following year the changes accour: F₁ keeps 85% of its consumers, loses 5% to F2 and 10% to F3 F2 keeps 55% of its consumers, loses 10% to F₁ and 35% to F3 F3 keeps 85% of its consumers, loses 10% to F₁ and 5% to F₂. Write the matrix S = the initial share and the matrix T = tij, the transformation matrix, where tij = what F; gets from F;. Compute TS, T· (T-S), T (T- (TS)). What do they represent?
(4) Three firms, F₁, F2, F3 share the market for a certain commodity. F₁ has 20%, F₂ has 60% and F3 has 20%. In the following year the changes accour: F₁ keeps 85% of its consumers, loses 5% to F2 and 10% to F3 F2 keeps 55% of its consumers, loses 10% to F₁ and 35% to F3 F3 keeps 85% of its consumers, loses 10% to F₁ and 5% to F₂. Write the matrix S = the initial share and the matrix T = tij, the transformation matrix, where tij = what F; gets from F;. Compute TS, T· (T-S), T (T- (TS)). What do they represent?
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 28EQ
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Transcribed Image Text:(4) Three firms, F₁, F2, F3 share the market for a certain commodity. F₁ has 20%, F2 has 60%
and F3 has 20%. In the following year the changes accour:
F₁ keeps 85% of its consumers, loses 5% to F2 and 10% to F3
F2 keeps 55% of its consumers, loses 10% to F₁ and 35% to F3
F3 keeps 85% of its consumers, loses 10% to F₁ and 5% to F2.
Write the matrix S = the initial share and the matrix T = tij, the transformation matrix,
where tij = what F; gets from Fj. Compute T - S,T· (TS), T · (T · (T· S)). What do they
represent?
(5) Calculate the following determinants:
3 -1 2
0 −1 -1
6 1 2
2-0 3 -1
0 4 0
0
0 1
-1
2
3 2
5
-3
30 1
1 0 -1
20 5
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