Wronskian to determine lin.ind. of solutions to a system of ODEs

[SithsNGiggles]

Homework Statement

In my book, I'm given that $\vec{x}_1=\left(\begin{matrix}t^2\\t\end{matrix}\right), \vec{x}_2=\left(\begin{matrix}0\\1+t\end{matrix}\right), \vec{x}_3=\left(\begin{matrix}-t^2\\1\end{matrix}\right)$ are solutions. My textbook presents an algebraic way to show that the vectors are linear independent, but I was hoping to see if I can use the Wronskian to show the same result.

Homework Equations

The Attempt at a Solution

I thought this was how the Wronskian would look:
$W\left(\vec{x}_1, \vec{x}_2, \vec{x}_3\right)= \left|\begin{matrix} \left(\begin{matrix}t^2\\1\end{matrix}\right) & \left(\begin{matrix}0\\1+t\end{matrix}\right) & \left(\begin{matrix}-t^2\\1\end{matrix}\right)\\ \left(\begin{matrix}2t\\0\end{matrix}\right) & \left(\begin{matrix}0\\1\end{matrix}\right) & \left(\begin{matrix}-2t\\0\end{matrix}\right)\\ \left(\begin{matrix}2\\0\end{matrix}\right) & \left(\begin{matrix}0\\0\end{matrix}\right) & \left(\begin{matrix}-2\\0\end{matrix}\right) \end{matrix}\right|$

But I couldn't see how to proceed from there, since matrix multiplication won't work. How would I find this determinant?

 

[vela]

You can't because it doesn't make sense.

In any case, you should be able to see by inspection that the three vectors aren't linearly independent.

 

[SithsNGiggles]

Okay, thanks.