Determine whether the following equations will have non-trivial solutions, if so solve them: `x+3y-2z=0, 2x-y+4z=0,x-11y+14z=0`

Explanation

We have, `x+3y-2z=0`
` 2x-y+4z=0`
`x-11y+14z=0`
The given system of equations in the matrix form are written as below.
`[(1,3,-2),(2,-1,4),(1,-11,14)][(,x),(,y),(,z)]=[(,0),(,0),(,0)]`
AX=O
where `A=[(1,3,-2),(2,-1,4),(1,-11,14)],X=[(,x),(,y),(,z)]and O=[(,0),(,0),(,0)]`
`therefore " " |A|=1(-14+44)-3(28-4)-2(-22+1)`
`=30-72+42=0`
and therefore the system has a non-trival solution, Now, we any write first two of the given equations
`x+3y=2z and 2x-y=-4z`
solving these equations in terms of z, we get
Putting `x=-(10)/(7)z-(8)/(7)z` in third equation of the given system,
we get , `LHS=-(10)/(7)z-(88)/(7)z=0=RHS`
Now, if `z=7k,then x=-10k and y=8k.`
Hence, `x=-10k,y=8k and z=7k` (where k is arbitary) are the requried solutions.