Determine all values of k for which the system

10 k -1 0

k 1 -1 0

2 1 -3 0

has nontrivial solution.

I have written the augmented matrix, with the solutions on the fourth row. My first instinct was to use Gaussian elimination to reduce the matrix to echelon form and peal off the solutions. However, this quickly became impossibly complex. Can anyone offer me any advice?