The question is related to the population of rabbits. It involves a matrix that shows the rate of change of the population of rabbits over time. The goal is to find the total population of rabbits at a specific point in time.


To calculate the total population of rabbits, we need to use matrix multiplication. The matrix given in the question represents the rate of change of the population over time. We can use this matrix to calculate the total population by multiplying it with a vector that represents the initial population of rabbits.


Let us define the following variables:

• A - Matrix that represents the rate of change of the population over time


• x - Vector that represents the initial population of rabbits


• t - Time at which we want to calculate the population

The formula to calculate the population at time t is given by:

P(t) = A^t * x

Where A^t represents A to the power of t (matrix multiplication of A with itself t times).


Suppose we have the following matrix A and vector x:

A = [0.8 0.2; 0.6 0.4]
x = [100; 50]

This means that at time t=0, there are 100 adult rabbits and 50 young rabbits.

To calculate the population at time t=3, we can use the formula:

P(3) = A^3 * x

Calculating A^3, we get:

A^3 = [0.688 0.312; 0.72 0.28]

Multiplying A^3 with x, we get:

P(3) = [81.6; 68.4]

This means that at time t=3, the total population of rabbits is 81.6 adult rabbits and 68.4 young rabbits. The total population is 150 rabbits.


In conclusion, the question involves finding the total population of rabbits at a specific point in time. This can be done by using the matrix given in the question, which represents the rate of change of the population over time. We can calculate the total population by multiplying this matrix with a vector that represents the initial population of rabbits. The formula for calculating the population at a specific time is P(t) = A^t * x.