How to identify if parameters are estimable, after defining a design matrix I need to define a design matrix $X$ so that it fits this scenario:
For $1\leq i< j \leq 5$, where $i$ are competitors and $j$ are different competitors. Their score is $y_{ij} =$ score of player $i$ minus score of player $j$.
$$Y_{ij} =\beta_i - \beta_j+\varepsilon_{ij}.$$
$\beta_i,\ldots,\beta_5$  are unknown parameters. $\varepsilon \sim N(0,\sigma^2I)$.
The $X$ matrix has to represent only 4 matches: $y_{12}$, $y_{34}$, $y_{25}$, $y_{15}$. and then be usable in general linear form $y=X\beta+\varepsilon$.
A couple challenges: 
(a) I've never done this before. I am thinking I should use zeros, ones and negative ones. Am I right in putting ones and negative ones in the two parameter columns for a match, and zeros in the parameter columns that aren't playing in the match?
(b) How do I decide if a couple comparisons can be made, $\beta_1 - \beta_2$ and $\beta_1 - \beta_3$?
$X =$ in the row order  $y_{12}$, $y_{34}$, $y_{25}$, $y_{15}$
$$
\begin{bmatrix}
1 &  -1  & 0 & 0 & 0 \\
0  & 0 & 1 & -1 & 0 \\
0 & 1 & 0 & 0 & -1 \\
1  &  0 & 0 & 0 & -1 \\
\end{bmatrix}
$$
I further have $E(y)$ = 
$$
\begin{bmatrix}
\mu_1 \\
\mu_2 \\
\mu_3 \\
\mu_4 \\
\end{bmatrix}
$$
I know of 2 ways to see if it's estimable. The first is by seeing if a matrix $A$ exists such that $Ae(Y) = AX\beta.$ The other way is the way I need to practice, so where I look to you: By seeing if there is a given linear combination of the elements of beta that can be written as a linear combination of the elements of E(y).
 A: The design matrix you have set up is not quite right. Since this is a homework problem, I'll give you hints and help you through the problem.  The key to setting up your design matrix involves, understanding what Y represents.  In this case, is represents, the difference between player $i$'s score and player $j$'s score.  So, to get started set up the matrices (in doing so, I'm ignoring the error terms $\epsilon_{ij}$, but remember those are important!) accordingly:
$\begin{bmatrix}
    y_{12}\\
    y_{34}\\
    y_{25}\\
    y_{15}\\
\end{bmatrix}=X\begin{bmatrix}
    \beta_1\\
    \beta_2\\
    \beta_3\\
    \beta_4\\
\end{bmatrix}=\begin{bmatrix}
    \beta_1-\beta_2\\
    \beta_3-\beta_4\\
    \beta_2-\beta_5\\
    \beta_1-\beta_5\\
\end{bmatrix}$
The $X$ matrix is a 4 $\times$5 matrix, where each column is represented by a $\beta$.  So you need to ask yourself, what should the elements of $X$ be so that when you multiply $X\beta$ you get the differences of $\beta$s shown on the right hand sight of the matrix equation above.  The elements should consists of only 1, -1, and 0.  Once you do this correction, it should be fairly straight forward to answer your remaining questions.
