Answer
False
Work Step by Step
To prove the statement, we obtain:
$x_1(t)=\begin{bmatrix}
x_{11}(t)\\
x_{12}(t)
\end{bmatrix}$
and $x_2(t)=\begin{bmatrix}
x_{21}(t)\\
x_{22}(t)
\end{bmatrix}$
then we have: $$W_{[x_1,x_2]}(t)=\begin{bmatrix}
x_{11}(t) & x_{21}(t)\\
x_{12}(t) & x_{22}(t)
\end{bmatrix}\\
=x_{11}(t)x_{22}(t)-x_{12}(t)x_{22}(t)$$
$$W_{[x_2,x_1]}(t)=\begin{bmatrix}
x_{21}(t) & x_{11}(t)\\
x_{22}(t) & x_{12}(t)
\end{bmatrix}\\
=x_{21}(t) x_{12}(t) -x_{22}(t) x_{11}(t) \\
=-[x_{11}(t) x_{22}(t)-x_{12}(t) x_{21}(t) ] $$
Hence, we can see that $W_{[x_2,x_1]}(t)=-W_{[x_1,x_2]}(t)$
