Help me spot the error? I have a determinant to expand which is 
$$\triangle = 
\begin{bmatrix}
 p& 1  & \frac{-q}{2}{}\\
 1& 2 &-q \\
 2& 2 & 3
\end{bmatrix} = 0 $$ 
But when I am expanding the determinant along the first row such as $ p(6+2q) - (3+2q)  = 0 $ but when I am trying to expand along first column I am getting $p(6+2q) - (3+q)  = 0$ but I have been told by my teacher that we can expand the determinant of $3\times 3$ matrix in along any row and any column giving the same result. where lies the error ? 

Seems you like you forgot the third column ($\frac{-q}{2}$) when expanding using the first row.
In the first row expansion, just using the first and second columns gives you $p(6+q) - (3 +2q)$. The third column gives an extra $q$, after adding which it matches the second expression you got, using the first column.
