1. Matrices pathway help

More matrix shenanigans. I find the matrix, but then it asks me to
" Write the matrix that represents the number of communication links with exactly one intermediary computer."
That's asking for me to do so with the initial matrix,

A B C D
A 0 1 1 0

B 1 0 1 0

C 1 0 0 1

D 0 0 0 0

How the heck do I turn the # of intermediaries into a matrix?

2. Hello, MathIsRelativelyUseless!

I hope I understand the problem . . .

Initial transition matrix: . $\displaystyle M \;=\;\begin{array}{c|cccc} & A & B & C & D \\ \hline A & 0 & 1 & 1 & 0 \\ B & 1 & 0 & 1 & 0 \\ C & 1 & 0 & 0 & 1 \\ D & 0 & 0 & 0 & 0 \end{array}$
The matrix show the possible transitions from one state to another in one step.

The element in the upper-left, $\displaystyle a_{11} \:=\:0$, represents $\displaystyle A \to A.$
. . This says: the process cannot go from state $\displaystyle A$ to state $\displaystyle A$ in one step.

The next element, $\displaystyle a_{12} \:=\:1$ represents $\displaystyle A \to B.$
. . This says: the process can go from state $\displaystyle A$ to state $\displaystyle B$ in one step.

And so on . . .

If there is one intermediary computer, we are dealing with a two-step process.
. . We want the matrix for all the possible two-step transitions.

This is a surprisingly simple process . . .

The two-step matrix is simply $\displaystyle M^2$

. . $\displaystyle M^2 \;=\;\begin{bmatrix}0 & 1 & 1 & 0 \\ 1 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{bmatrix}\begin{bmatrix}0 & 1&1&0\\1&0&1&0\\1&0&0&1\\0&0&0&0 \end{bmatrix} \;=\;\begin{bmatrix}2&0&1&1\\1&1&1&1\\0&1&1&0\\0&0 &0&0\end{bmatrix}$

This can be read as another chart: . $\displaystyle \begin{array}{c|cccc}M^2 & A & B & C & D \\ \hline A & 2 & 0 & 1 & 1 \\ B & 1 & 1 & 1 & 1 \\ C & 0 & 1 & 1 & 0 \\ D & 0 & 0 & 0 & 0 \end{array}$

The upper-left element, $\displaystyle a_{11} = 2$, represents $\displaystyle A \to A$ in two steps.
. . The process can go from $\displaystyle A$ to $\displaystyle A$ in two steps (in two ways).

The next element, $\displaystyle a_{12} = 0$, represents $\displaystyle A \to B$ in two steps.
. . The process cannot go from $\displaystyle A$ to $\displaystyle B$ in two steps.

And so on . . .

And yes, for a three-step process, determine $\displaystyle M^3$ . . .

3. Thank you sir!

That is so much more than what the book explained...

4. Ah, now I am being asked,

And I need to find P1 and P3.

What?
WHAT?
Why are all of the questions I'm being asked ones that weren't even touched upon by the crappy online course NOR the textbook? What a poorly put together course this is...

Now if you people here taught a class, then we'd be somewhere

5. Uhh....not to be hasty..but...anybody? The assignment's already late because I have no idea how to finish this, lol.

6. Originally Posted by MathIsRelativelyUseless
Why are all of the questions I'm being asked ones that weren't even touched upon by the crappy online course NOR the textbook?
Part of your problems comes from an online course.
If I had my way there would be no online mathematics courses.

Given $\displaystyle P_0 = \left( {\begin{array}{lll} {0.2} & {0.5} & {0.3} \\ \end{array} } \right)\,\& \,T = \left[ {\begin{array}{ccc} {0.2} & {0.2} & {0.6} \\ {0.5} & 0 & {0.5} \\ {0.6} & {0.3} & {0.1} \\ \end{array} } \right]$.

These problems are about Markov chains.
$\displaystyle P_1 = P_0 \cdot T = \left( {\begin{array}{lll} {0.47} & {0.13} & {0.4} \\ \end{array} } \right)$ That is simple matrix multiplication.

$\displaystyle P_2 = \left( {P_0 \cdot T} \right) \cdot T = P_0 \cdot T^2 = \left( {\begin{array}{*{20}c} {0.399} & {0.214} & {0.387} \\ \end{array} } \right)$

In general: $\displaystyle P_n = P_0 \cdot T^n$.

7. Thank you sir!

I just wrote the lame little online quiz (I hate taking this online, is makes it so much more difficult) and you helped with that a lot.