# Matrix problem

#### lorica2000

The Goal Soap Company sells laundry detergent in two - litre and five-litre packages. Their research shows that 34% of the people buying the small package will switch to the large package for their next purches, and 12% of the buyers of the large package will switch to the small package for the next purchase. the original market share was 55 % for the small package and 45% for the large package.

a. Write the initial population matrix for the number of buyers of large and small packages.
b. write the transition matrix that describes the switching behaviour of the buyers.
c. what is the market share for each size for the next round of purchase?
d. what is the market share for each size after 5 rounds of purchases?

Thanks

#### pickslides

MHF Helper
Here's a hint for your system, L denotes large packet, S denotes small

$$\displaystyle \begin{pmatrix}L_{n+1} \\ S_{n+1}\end{pmatrix} = \begin{pmatrix} a & 0.34 \\ 0.12 & b\end{pmatrix} \times \begin{pmatrix}L_{n} \\ S_{n}\end{pmatrix}$$

lorica2000

#### HallsofIvy

MHF Helper
The Goal Soap Company sells laundry detergent in two - litre and five-litre packages. Their research shows that 34% of the people buying the small package will switch to the large package for their next purches, and 12% of the buyers of the large package will switch to the small package for the next purchase. the original market share was 55 % for the small package and 45% for the large package.

a. Write the initial population matrix for the number of buyers of large and small packages.
b. write the transition matrix that describes the switching behaviour of the buyers.
c. what is the market share for each size for the next round of purchase?
d. what is the market share for each size after 5 rounds of purchases?

Thanks
Hey, loricao2000, are you and ecdino2 taking the same course?

Exactly what these matrices look like depends on how you choose to represent the population matrix. You will want to use a column matrix but you can do it either as
$$\displaystyle \begin{pmatrix}buyers\_of\_large\_packages \\ buyers\_of\_small\_packages\end{pmatrix}$$
or as
$$\displaystyle \begin{pmatrix}buyers\_of\_small\_packages \\ buyers\_of\_small\_large\end{pmatrix}$$

That should make it clear how to do (a).

For (b), you want a 2 by 2 matrix where each row shows how many buyers of either kind of package change into buyers of the kind of package the first row of your answer to (a) represent. That is, if you chose the first way to represent buyers of packages, the first column of your matrix will be $$\displaystyle begin{pmatrix}.88 \\ .34\end{pmatrix}$$ since both of those number will be multiplying the first row of the column matrix and 100-12= 88% of those buying the large package will stay with the large package and 34% will change to the small package.

For (c) and (d), multiply those matrices!

(I see that pickslides beat me again!)

Last edited:

thanks