gaussian elimination question, math?

Question

I tried finding solving this matrix but was only able to come up with the zero vector, and I'm pretty sure it does have a non trivial solution

row 1: ( 0 -1 0 ) = 0 row 2: (1 1 1) =0 row 3: ( 1 10 1) =0

the matrix is arranged with the rows on top of eachother

please help

Answer 1

arrange it like this:

[ 1 10 1 0]
[ 1 1 1 0 ]
[ 0 -1 0 0 ]

add 10 times row 3 to row 1 and row 3 to row 2:

[ 1 0 1 0]
[ 1 0 1 0 ]
[ 0 -1 0 0 ]

multiply row 3 by -1, and subtract row 1 from row 2:

[ 1 0 1 0]
[ 0 0 0 0 ]
[ 0 1 0 0 ]

swap row 3 and row 2:

[ 1 0 1 0]
[ 0 1 0 0 ]
[ 0 0 0 0 ]

you don't have 3 independent vectors. you have an infinite set of solutions of the form (x,0,-x), a line in the xz plane.