I have the vectors $A=a\hat e\_x$ and $B=a\hat e\_y$, so 
$$
A\times B =
\begin{vmatrix}
\hat e\_x &\hat e\_y & \hat e\_z \\
a & 0 & 0\\
0 & a &0
\end{vmatrix}=\hat za^2
$$
*Q1: But why is the following wrong*
\begin{align}
A\times B &=a\hat e\_x\times a\hat e\_y\\
&=(a\hat e\_x)\times (a\hat e\_y) \tag 1\\ 
&=(a\hat e\_x)\times (\hat e\_ya) \tag 2\\
&=a(\hat e\_x)\times (\hat e\_y)a \tag 3\\
&=a((\hat e\_x)\times (\hat e\_y)) \tag 4\\
&=a(\hat e\_x\times \hat e\_y) \tag 5\\
&=a\hat e\_z \quad ?
\end{align}

*Q2: Also, why is this wrong*
\begin{align}
A\times B &=
\begin{vmatrix}
\hat e\_x &\hat e\_y & \hat e\_z \\
a & 0 & 0\\
0 & a &0
\end{vmatrix} \\
&=
a
\begin{vmatrix}
\hat e\_x &\hat e\_y & \hat e\_z \\
1 & 0 & 0\\ \tag 6
0 & 1 &0
\end{vmatrix}\\
&=a\hat z \quad ?
\end{align}