Chapter 2.4, Problem 29E

(a)

To determine

The matrix equation relating $\left[\begin{array}{c}x\\ y\end{array}\right]$ to $\left[\begin{array}{c}u\\ v\end{array}\right]$ if x and y denotes the initial number of organisms in group I and II respectively and a and b denotes their corresponding numbers of organisms in group I and II respectively after one year such that there are two groups of organism in which organisms up to $1$ years of age are categorized under group I and organisms from $1$ to $2$ years of age are categorized under group II. Nine-tenths of group I survive to enter group II each year.

(b)

To determine

To calculate: The number of organisms in each group after $1$ year and after $2$ years if there are initially $450,000$ in group I and $360,000$ in group II where x and y denotes the initial number of organisms in group I and II respectively and a and b denotes their corresponding numbers of organisms in group I and II respectively after one year such that there are two groups of organism in which organisms up to $1$ years of age are categorized under group I and organisms from $1$ to $2$ years of age are categorized under group II. Nine-tenths of group I survive to enter group II each year.

(c)

To determine

To calculate: The number of organisms in each group before $1$ year if there are initially $810,000$ in group I and $630,000$ in group II where x and y denotes the initial number of organisms in group I and II respectively and a and b denotes their corresponding numbers of organisms in group I and II respectively after one year such that there are two groups of organism in which organisms up to $1$ years of age are categorized under group I and organisms from $1$ to $2$ years of age are categorized under group II. Nine-tenths of group I survive to enter group II each year.