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in Matrices & determinants by (15 points)
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The system of equation \( 2 x+3 y-z=0,3 x+2 y+k z=0,4 x+ \) \( y+z=0 \) have a set of nonzero integral solution then the least positive value of \( z \) is

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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.

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