<<VilGebra`

Solve[Det[{{k,2,3,1},{k,-k,1,1+k},{k,k,0,0},{1,1,0,1-k}}]==0,k]

En k=-2/5:

{{k,2,3,1,k},{k,-k,1,1+k,2},{k,k,0,0,0},{1,1,0,1-k,1-k}}/.k->-2/5//MatrixForm
MatrixForm[RowReduce[%]]

En k=0:

{{k,2,3,1,k},{k,-k,1,1+k,2},{k,k,0,0,0},{1,1,0,1-k,1-k}}/.k->0//MatrixForm
MatrixForm[RowReduce[%]]

En k=1:

{{k,2,3,1,k},{k,-k,1,1+k,2},{k,k,0,0,0},{1,1,0,1-k,1-k}}/.k->1//MatrixForm
MatrixForm[RowReduce[%]]