Question 1015937
Look at the coefficient matrix,
{{{A=(matrix(3,3,
k,1,1,
1,k,1,
1,1,k))}}}
Find the determinant,
{{{D=k(k^2-1)+1(1-k)+1(1-k)}}}
{{{D=k^3-k+2-2k}}}
{{{D=k^3-3k+2}}}
{{{D=(k-1)^2(k+2)}}}
So {{{k=1}}} and {{{k=-2}}} are solutions.
When {{{k=1}}},
{{{A=(matrix(3,3,
1,1,1,
1,1,1,
1,1,1))}}}
and the right hand side becomes,
{{{b=(matrix(3,1,3,3,3))}}}
So this system is dependent and has infinitely many solutions.
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When {{{k=-2}}},
{{{A=(matrix(3,3,
-2,1,1,
1,-2,1,
1,1,-2))}}}
and the right hand side becomes,
{{{b=(matrix(3,1,6,0,3))}}}.
In this case, it is an inconsistent system because {{{abs(A[x])<>0}}}, 
{{{abs(A[y])<>0}}}, and {{{abs(A[z])<>0}}}, 
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For all other values of k, you would have a unique solution.