Suppose that we have a group of six people. each of whom

Chapter , Problem 26

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Suppose that we have a group of six people. each of whom own< a communication device. We define a 6 x 6 matrix A a; follows: For I !5 i !5 6. let a, = 0: and for i -# j. { I if person i can send a message to person j a,, = 0 othen,ie. SOLUTIONS TO THE PRACTICE PROBLEMS I. (a) A = [~:;~ ~l (b) The population d1stnbution of females for this year is ghcn by the vector [~:]. and hence the population distributio n of females for next year is [ 200] [0.50 2] [ 200] [500] A 200 = 0.25 0 200 = 50 . and the population distibution of females 2 years from now is [ 500] = [0.50 A 50 0.25 2] [500] [350] 0 50 = 125 . (c) Since A [400] _ [0.50 100 - 0.25 2.2 Applications of Matrix Multiplication 121 (a) Show that A is a (0. I )-matrix. (b) Give an interpretatiOn of what it means for the term (IJ2(/2J to equal one. (c) Show that the (3. I )-entry of A 2 represents the number of ways that person 3 can send a mc~sage to person I in two stag. the number of people to whom person 3 can ;end a mes.agc and "ho in tum can send a message to person I. flim: Con~ider the number of terms that arc not equal to zero in the expression (d) GencraliLc your rc,ult in (c) to the (i.j)-cntry of A2 (c) Generalize your rc'ult in (d) to the (i.j)entry of A'". Now 'uppo'e 0 0 I I 0 0 I 0 0 0 0 () 0 ll (f) Is there uny person who c:~nnot receive ll mc.-sagc from anyone else in one >luge? Justify your :~nswer. (g) How many way' can pcr..on I ;end a mt:tgc to per son 4 in I, 2. 3, and 4 sl:lges? (h) The (i.j)-entry of A +1\ 2 + .. +A"' can be shown to equal the number of ways in which person i can send a message to person J in at most m stages. Use this result to dctcrmmc the number of ways in which person 3 can send a message to person 4 in at most 4 tage.~.