Multiplying Adjacent Matrices? My teacher hasn't explained it too well, so i'm looking for an explanation:
$$A =
\begin{pmatrix}
  0 & 1 & 1\\
  1 & 0 & 1\\
  1 & 1 & 0
\end{pmatrix}$$
$$A^2 =
\begin{pmatrix}
  2 & 1 & 1\\
  1 & 2 & 1\\
  1 & 1 & 2
\end{pmatrix}$$
I would have thought the first number in $A^2$ would be $0$, as $0 \cdot 0 = 0$, and am struggling to see where the $2$ comes from. 

$A^2$ is simply $A\cdot A$, where the $\cdot$ denotes matrix multiplication.
Matrices are not multiplied elementwise (other than for addition), but as follows:
The matrix element of the resulting matrix which sits on row $i$ and column $j$ is calculated by multiplying each element of $i$-th row of the first matrix with the corresponding element of the $j$-th column of the second matrix, and add up all those products. For your specific case, when interested in the upper left element of
$$\begin{pmatrix}0&1&1\\1&0&1\\1&1&0\end{pmatrix}\cdot\begin{pmatrix}0&1&1\\1&0&1\\1&1&0\end{pmatrix}$$
you have to look at the first row of the first matrix,
$$\begin{pmatrix}0&1&1\end{pmatrix}$$
and the first column of the second matrix,
$$\begin{pmatrix}0\\1\\1\end{pmatrix}$$
and then find $0\cdot 0+1\cdot 1+1\cdot 1 = 2$. Therefore the upper left matrix element of the resulting matrix is $2$.
