I've been struggling with this problem for a while.

find values of k so that the system in unknowns x y and z has a unique solution, no solution, infinite solution

kx + y + z = 1

x + ky + z = 1

x + y + kz = 1

I know that the forms:

0 = 0 is an infinite solution

0 = 1 is no solution

nonzero number = other nonzero number is a unique solution