1. ## matrix problem

An airline company offers daily flights among Budapest, Frankfurt, Tunis, and Windsor
The number of daily direct flights is as follow:

From Windsor - 2 flights to Tunis, one flight to Budapest and one flight to Frankfurt
From Budapest there are 2 flights to Windsor
From Tunis ther is one flight to Windsor and one to Frankfurt
From Frankfurt there are two flights to Windsor and one to Tunis

I've found the matrix that indicate the number of routes between cities which is
W B F T
W 0 1 1 2
B 2 0 0 0
F 2 0 0 1
T 1 0 1 0

My question is - indicate the matrix operation that show the number of ways that a plane can go between two cities, either directly or through at most one city

R + Rsquared (I've been told) but I don't understand why. I need some feedback.

Thanks

2. Originally Posted by sdinulescu
An airline company offers daily flights among Budapest, Frankfurt, Tunis, and Windsor
The number of daily direct flights is as follow:

From Windsor - 2 flights to Tunis, one flight to Budapest and one flight to Frankfurt
From Budapest there are 2 flights to Windsor
From Tunis there is one flight to Windsor and one to Frankfurt
From Frankfurt there are two flights to Windsor and one to Tunis

I've found the matrix that indicate the number of routes between cities which is
WjW B F T
WX0 1 1 2
BX2 0 0 0
FX2 0 0 1
TX1 0 1 0

My question is - indicate the matrix operation that show the number of ways that a plane can go between two cities, either directly or through at most one city

R + Rsquared (I've been told) but I don't understand why. I need some feedback.
Each entry in the matrix $R$ tells you the number of ways of going directly from the "row" city to the "column" city. Each element of $R^2$ tells you the number of ways of getting from the "row" city to the "column" city via an intermediate city. For example, if $R_{XY}$ denotes the element of $R$ in row X and column Y, then the WF-element of $R^2$ is given (according to the rule for matrix multiplication) by $R^2_{WF} = R_{WW}R_{WF} + R_{WB}R_{BF} + R_{WF}R_{FF} + R_{WT}R_{TF}.$ The right side of that expression is the number of ways of getting from W to F via W, B, F and T respectively.

So if you want the number of ways to go between two cities, either directly or with one stopover, you add the number of direct routes to the number with one stopover, and the matrix for that will be $R+R^2.$