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