Efficient Method for Finding Basis and Determinant of 4 Vectors in Matrix Form

[pyroknife]

I know how to do the problem, just put the 4 vectors in matrix form and find for what values of k is the detminant =0. the answer is then that k can't equal the value that was found.

Is there a easier way to do this?

My method involves finding the determinant using the expansion method, which seems like a long way. Is there a faster way?

 

[haruspex]

In this example, it's quicker to spot simplifications first.
Add 3rd vector to 2nd to produce (0, 2, 0, 7).
Multiply 1st by 7 and new 2nd by 2 then subtract 2nd from 1st.
etc.