At an exclusive retreat for wealthy oil company executives, it is found that the

Question

At an exclusive retreat for wealthy oil company executives,
it is found that there are exactly three types of executives attending:
i) those that own exactly 8 homes, 7 cars, and 7 dogs
ii) those that own exactly 4 homes, 3 cars, and 5 dogs
iii) and the poor souls who own exactly 2 homes, 2 cars and 1 dog
Suppose you are told the total number of homes and cars owned by the oil executives.
express the total number of dogs, D, in terms of the total
number of houses, H, and the total number of cars, C. Are the three totals independent
pieces of information?

Explanation / Answer

Let there are x1,x2,x3 number of type1, 2 and 3 executives repectively attending the meeting. H=total number of homes =8x1+4x2+2x3 C=total number of cars=7x1+3x2+2x3 D=total number of dogs=7x1+5x2+x3 By gaussian elimination from the system of equations 8 4 2 H 7 3 2 C 7 5 1 D = 8 4 2 H 0 -1/2 1/4 C-7*H/8 0 3/2 -3/4 D-7*H/8 = 8 4 2 H 0 -1/2 1/4 C-7*H/8 0 0 0 D-7*H/8+ (3/4)*(C-7*H/8) So D=(7/4)*(7/8)*H-(3/4)*C=(49/32)*H -(3/4)*C Clearly the three totals aren't independed informations as the system of equations aren't independent. The corresponding matrix is not full rank matrix.( it has rank 2)

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