The system of linear equations in three variables is given by
(k−1)x+y+z=1,
x+(k−1)y+z=1,
x+y+(k−1)z=1,
where k is a real parameter.
(i) Show that ∣A∣=(k−2)2(k+1), where A is the coefficient matrix of the system.
(ii) Using the result of part (i), determine the values of k for which the system has
(a) a unique solution,
(b) no solution, and
(c) infinitely many solutions.
(iii) For each value of k that yields a unique solution, obtain the solution (x,y,z).