The following LaTeX code looks pretty straightforward to me:
\begin{eqnarray*}
& \lambda_l \mathbf{l}\cdot\mathbf{r} & = & \lambda_r \mathbf{l}\cdot\mathbf{r} \nonumber \\
\therefore & \mathbf{l}\cdot\mathbf{r} & = & 0 \nonumber
\end{eqnarray*}
but it results in the error:
! Missing $ inserted.
<inserted text>
$
l.78 ..._l \mathbf{l}\cdot\mathbf{r} & = & \lambda
_r \mathbf{l}\cdot\mathbf{...
If I remove enough ampersands, I get a clean execution, but then I lose the alignment I am trying to get.
Is there a problem with the implicit first empty item on the first line, or is it caused by something else?

&per line. On the other hand,eqnarrayis not the right tool for the job. – egreg Apr 11 '17 at 14:38eqnarrayto begin with: it's broken; seeeqnarrayvsalign– egreg Apr 11 '17 at 14:44eqnarray, i never thought of it as "broken" to the extent that it would produce such a misleading error message. now i agree -- it's definitely broken! – barbara beeton Apr 11 '17 at 17:49