What Values of k Ensure Linear Independence in R^4 for These Vectors?

[Nathew]

Homework Statement

Determine all values of the constant k for which the given set of vectors is linearly independent in $\mathbb R^4$.
{(1, 1, 0, −1), (1, k, 1, 1), (4, 1, k, 1), (−1, 1, 1, k)}

Homework Equations

The Attempt at a Solution

So far I set up a coefficient matrix
$$\begin{pmatrix}<br /> 1 & 1 & 4 & -1 \\<br /> 1 & k & 1 & 1 \\<br /> 0 & 1 & k & 1 \\<br /> -1 & 1 & 1 & k<br /> \end{pmatrix}$$

And tried converting it to REF

$$\begin{pmatrix}<br /> 1 & 1 & 4 & -1 \\<br /> 0 & 1 & k & 1 \\<br /> 0 & 0 & (-k^2+k-3) & (3-k) \\<br /> 0 & 0 & (5-2k) & (k-3)<br /> \end{pmatrix}$$

I'm not sure if I should keep going trying to reduce this to REF to see which values of k will not work, but it just seems too messy.

Am I approaching this the wrong way?

 

[LCKurtz]

Unlikely as it may seem, it looks to me like checking the determinant is as easy or easier than row reduction. If you add the last row to each of the first two rows you get a column with 3 zeros leaving you with one 3x3 determinant, which is easy to just expand.