Consider the system of equations

x + 2y + 3z = 0

2x y = 0

x + kz = 0.

a) Calculate the determinant of the coefficient matrix using row operations.

b) For which value(s) of k (if any) does the system have no solution?

c) For which value(s) of k (if any) does the system have exactly one solution?

d) For which value(s) of k (if any) does the system have infinitely many solutions?