Take the determinant of your coefficient matrix and set it equal to zero. You should get a cubic polynomial in k, and get up to three solutions.

For all values of k other than those (up to) three solutions, the determinant is non-zero and the matrix is non-singular. If this is the case, then there is always a unique solution, no matter what the right hand side is.

For k which make the determinant equal to zero, the matrix is singular. This means that there is always either no solution or infinitely many solutions. Since there are at most three such values of k, you can test each one by hand to see which is the case.

For example, k = 1 gives a zero determinant. For k = 1, all three equations are identical and you get infinitely many solutions.