The given system has a set of non zero integral solution and system is homogeneous system.
\(\therefore \begin{vmatrix}2&3&-1\\3&2&k\\4&1&1 \end{vmatrix} \ne 0\)
⇒ \(2(2 - k) - 3(3 - 4k) -1(3 - 8) \ne 0\)
⇒ \(4 - 2k -9 + 12k + 5 \ne 0 \)
⇒ \(10k \ne 0\)
⇒ \(k\ne 0\)
System has integral solution, so k must have integral values.
Hence, least positive value of z is 1.