Answer
312 cars the airport, 58 cars are the east side office, and 130 cars at the west side office
Work Step by Step
Remember that the difference equation describing migrations of this example can be considered a migration of cars is
\[
\mathbf{x}_{k+1}=M \mathbf{x}_{k}
\]
where in this case, the migration matrix $M$ is given.
We know that if we construct a 'population vector' $\mathbf{x}_{0}=\left[\begin{array}{c}295 \\ 55 \\ 150\end{array}\right]$ describing the quantity of cars in each of the three locations on Monday, then we will get the populations on Tuesday by evaluating
\[
M \mathbf{x}_{0}
\]
To get the population vector for Wednesday, we need to apply the migration matrix to this vector:
\[
M\left(M \mathbf{x}_{0}\right)
\]
This is easy to evaluate using a matrix software. The following syntax will work in either MATLAB or Octave:
\[
\text { octave: } 1>M=\left[\begin{array}{llll}
\theta & 97 & , & \theta & 05
\end{array}, 0.1 ; 0,0.9,0.05 ; 0.03,0.05,0.85\right] ;
\]
octave: $2>x \theta=[295 ; 55 ; 15 \theta]$
octave: $3>M *(M * x \theta)$ ans $=$
\[
\begin{array}{r}
311.543 \\
58.255 \\
130.202
\end{array}
\]
There are unlikely to be cars returned in pieces $;-),$ but we should have a good estimate by rounding the numbers found above.
