Can the loops be in any order in the Floyd-Warshall algorithm?
Suppose that you change the order of loops from $k,i,j$ to $i,j,k$. Let's see what happens when $i=1$ and $j=2$:
for k from 1 to |V|
  if dist[1][2] &gt; dist[1][k] + dist[k][2]
    dist[1][2] ← dist[1][k] + dist[k][2]
  end if
end for

It is not too hard to check that this puts in dist[1][2] the value
$$
\min_{1 \leq k \leq |V|} w(1,k) + w(k,2),
$$
where $w(1,1) = w(2,2) = 0$. This is the cost of the shortest path of length at most $2$ from $1$ to $2$.
The $(1,2)$'th iteration is the only one which modifies dist[1][2], so at the end of the procedure, dist[1][2] will still contain
$$
\min_{1 \leq k \leq |V|} w(1,k) + w(k,2),
$$
which could be arbitrarily off from the real shortest path from $1$ to $2$.