Three clients of Disrupt, Ltd - P, Q, and R - are direct competitors in the retail business. In the first week of the year, P had 300 customers, Q had 250 customers, and R had 200 customers. During the second week, 60 of the original customers of P transferred to Q and 30 of the original customers of P transferred to R. Similarly, in the second week, 50 of the original customers of Q transferred to P with no transfers to R, and 20 of the original customers of R transferred to P with no transfers to Q.
Required:
a) Display in a matrix the pattern of retention and transfers of customers from the first to the second week.
b) Re-express the matrix that you have obtained in part (a) showing the elements as decimal fractions of the original numbers of customers of P, Q, and R.
c) Multiply matrix B by itself to determine the proportions of the original customers that have been retained or transferred to P, Q, and R from the second to the third week.
d) Solve the matrix equation (xyz)B = (xyz) given that x + y + z = 1.
e) Interpret the result that you obtain in part (d) in relation to the movement of customers between P, Q, and R.