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. 

The first entry in $A^2$ comes from the first entry in $AA$, and that entry is $0\cdot 0 + 1\cdot 1 + 1 \cdot 1 = 2$.
Likewise, the entry in the first row, second column of $A^2$ is $0\cdot 1 + 1\cdot 0 + 1 \cdot 1 = 1$.
Perhaps you'll want to revisit matrix multiplication.
In particular, Wikipedia provides this example for arbitrary $3\times 3$ matricies:
Square matrices
If
$$\mathbf{A} = \begin{pmatrix} a & b & c \\ p & q & r \\ u & v & w \end{pmatrix}, \quad \mathbf{B} = \begin{pmatrix} \alpha & \beta & \gamma \\ \lambda & \mu & \nu \\ \rho & \sigma & \tau \\ \end{pmatrix}\,,$$
their matrix products are:
$$\mathbf{AB} = \begin{pmatrix} a & b & c \\ p & q & r \\ u & v & w \end{pmatrix} \begin{pmatrix} \alpha & \beta & \gamma \\ \lambda & \mu & \nu \\ \rho & \sigma & \tau \\ \end{pmatrix} =\begin{pmatrix} a\alpha + b\lambda + c\rho & a\beta + b\mu + c\sigma & a\gamma + b\nu + c\tau \\ p\alpha + q\lambda + r\rho & p\beta + q\mu + r\sigma & p\gamma + q\nu + r\tau \\ u\alpha + v\lambda + w\rho & u\beta + v\mu + w\sigma & u\gamma + v\nu + w\tau \end{pmatrix}\,,$$ 
and
$$\mathbf{BA} = \begin{pmatrix} \alpha & \beta & \gamma \\ \lambda & \mu & \nu \\ \rho & \sigma & \tau \\ \end{pmatrix} \begin{pmatrix} a & b & c \\ p & q & r \\ u & v & w \end{pmatrix} =\begin{pmatrix} \alpha a + \beta p + \gamma u & \alpha b + \beta q + \gamma v & \alpha c + \beta r + \gamma w \\ \lambda a + \mu p + \nu u & \lambda b + \mu q + \nu v & \lambda c + \mu r + \nu w \\ \rho a + \sigma p + \tau u & \rho b + \sigma q + \tau v & \rho c + \sigma r + \tau w \end{pmatrix}\,.$$ 
