In $\mathbb{R}$, we have the notion of multiplying an element by $-1$. For example:
$12 \in \mathbb{R}$, then $12 \times (-1) = -12 \in \mathbb{R}$,
$7 \in \mathbb{R}$, then $7 \times (-1) = -7 \in \mathbb{R}$.
Now, assume we are in $GF(4)$ and its definition is a standard definition with elements $\{0, 1, \omega, \bar{\omega}\}$. We will label them as $\{0, 1, 2, 3\}$, respectively:
$$ \begin{array}{|c|cccc|}\hline + & 0 & 1 & 2 & 3\\\hline 0 & 0 & 1 & 2 & 3\\ 1 & 1 & 0 & 3 & 2\\ 2 & 2 & 3 & 0 & 1\\ 3 & 3 & 2 & 1 & 0\\\hline \end{array}\qquad \begin{array}{|c|cccc|}\hline - & 0 & 1 & 2 & 3\\\hline 0 & 0 & 1 & 2 & 3\\ 1 & 1 & 0 & 3 & 2\\ 2 & 2 & 3 & 0 & 1\\ 3 & 3 & 2 & 1 & 0\\\hline \end{array}\qquad \begin{array}{|c|cccc|}\hline \times & 0 & 1 & 2 & 3\\\hline 0 & 0 & 0 & 0 & 0\\ 1 & 0 & 1 & 2 & 3\\ 2 & 0 & 2 & 3 & 1\\ 3 & 0 & 3 & 1 & 2\\\hline \end{array}\qquad \begin{array}{|c|cccc|}\hline \div& 0 & 1 & 2 & 3\\\hline 0 & - & 0 & 0 & 0\\ 1 & - & 1 & 3 & 2\\ 2 & - & 2 & 1 & 3\\ 3 & - & 3 & 2 & 1\\\hline \end{array}\qquad $$
It is usually understood that $-x$ is the additive inverse so that for any element $x \in GF(4)\,\,$, $x + (-x) = 0$.
I am interested in multiplying by $-1$ in $GF(4)$ defined as the same mathematical meaning as ordinary multiplying by $-1$ in $\mathbb{R}$. How would I go about doing something like this?
In the case where you would like more context, this deals with finding determinant of a matrix with elements in $GF(4)$ and multiplying by $-1$ is a necessary condition.
$${GF(4)= \{0,1,w,w^2\}}$$ Now, ${w+1=w^2}$, so ${w^3=w \cdot w^2=w \cdot(1+w)=w+w^2=w + (1+w)=1}$, so ${w^2}$ is multiplicative inverse of ${w}$.