Answer
The given system has an infinite number of solutions if and only if $k=4$ or $k = \pm 1$
Work Step by Step
The system can be written as:
$A=\begin{bmatrix}
1-k &1 &1\\
2 &1-k&1\\
1 & 1 & 2-k
\end{bmatrix}$
The determinant for this system is:
$\rightarrow \det A=(1-k)(1-k)(2-k)+2.1.1+1.2.1$
$=-k^3+4k^2+k-4$
The given system has an infinite number of solutions if and only if
$\det A =0$
$-k^3+4k^2+k-4= 0$
$(k-4)(-k^2+1)=0$
$k=4$ or $k = \pm 1$
