Matrix multiplication of $4 \times 1$ matrix with $4 \times 4$ matrix I'm new to matrix multiplication and just wondered how I would evaluate the following:
$$
\begin{pmatrix}
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 \\
0 & -1& 0 & 0 \\
-1& 0 & 0 & 0
\end{pmatrix}
\begin{pmatrix}
a \\
b \\
c \\
d \\
\end{pmatrix}
$$
and even:
$$
\begin{pmatrix}
a & b & c & d
\end{pmatrix}
\begin{pmatrix}
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 \\
0 & -1& 0 & 0 \\
-1& 0 & 0 & 0
\end{pmatrix}
$$

There is a formula to compute the $(i, j)$th entry of $C = AB$ (provided the sizes of $A$ and $B$ match): assume $A$ is $n \times p$ and $B$ is $p \times m$, then $C$ is $n \times m$, and 
$$c_{i, j} = \sum_{k = 1}^p a_{\color{red}{i}k}b_{k\color{red}{j}}.$$
Take your first question for example, the result should be a $4 \times 1$ matrix, it is quite straightforward to compute as follows:
$$
\begin{pmatrix}
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 \\
0 & -1& 0 & 0 \\
-1& 0 & 0 & 0
\end{pmatrix}
\begin{pmatrix}
a \\
b \\
c \\
d \\
\end{pmatrix}
= \begin{pmatrix}
0 \times a + 0 \times b + 0 \times c + 1 \times d \\
0 \times a + 0 \times b + 1 \times c + 0 \times d \\
0 \times a + (-1) \times b + 0 \times c + 0 \times d \\
(-1) \times a + 0 \times b + 0 \times c + 0 \times d 
\end{pmatrix}
= \begin{pmatrix}
d \\
c \\
-b \\
-a
\end{pmatrix}.
$$
