How Can Matrix Theory Predict Subscription Trends in a Concert Series?

[LakeMountD]

in a community of 100,000 adults, subscribers to a concert series tend to renew their subscription with a probability 90% and persons not subscribing will subscribe for the next season with probability 0.2%. If the present number of subscribers is 1200, can one predict an increase or decrease or stability over each of the next three seasons?

This is in the matrix section so I am assuming we have to use one but have no idea what they want?!

 

[HallsofIvy]

If you write the number of people who subscribed last year as X and the number of people who did not as Y, then the expected number of people to subscribe this year is 0.90X+ 0.002Y. Of course, the number of people who will not subscribe this year will be 0.10X+ 0.998Y. You can write that as a matrix: What matrix multiplied by the column vector (X,Y) will give the vector (0.90X+ 0.002Y, 0.10X+ 0.998Y)?

The easiest way to answer that question is to look at the determinant of the matrix.

 

[LakeMountD]

determinent of the matrix would be

[ .9 * .998] - [.002 * .10] correct?

 

[HallsofIvy]

Yes. What is that equal to and what does it tell you?

 

[LakeMountD]

= .898 .. not really sure what that is telling me though. honestly my differential equations teacher is so bad that i have to learn everything on google. pretty sad when your 5,000 dollars in tuition is going towards research you have to do on your own on google to learn everything but hey.

im assuming that since .898 is so close to .90 or 90% that its saying things are going to be stable. This right?

also how do you get a vector out of those numbers?