For what value of k does the system have a non-trivial solution? kx + 3y - 4z = 0   x +  y        = 0 3x + ky + 2z = 0  

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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Notice that the right column of terms consists of zeroes, hence, the system is called homogenous.

If the system has an unique solution, this solution is trivial solution, meaning that `x=0,y=0,z=0` .

The system has an unique solution if the determinant of matrix, `Delta` , of system has a value different form zero.

Since the problem provides the information that the system has non-trivial solution, hence determinant of matrix of system is zero such that:

`Delta = [[k,3,-4],[1,1,0],[3,k,2]]`

`Delta = 2k - 4k + 0 + 12 - 0 - 6`

But `Delta = 0 => 2k - 4k + 0 + 12 - 0 - 6 = 0`

-`2k + 6 = 0 => -2k = -6 => k = 3`

Hence, evaluating the value of k for the system to have non-trivial solution yields k = 3.

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