Detemine whether vector are linearly dependent

The vectors are (1,-2,3) (-1,3,2) and (-1,10,5) . Question asks to find if they are linearly independent.

So I've put them into matrix form, as below

now start gauss-jordon elimination.

new row2 = old row2 + 2*row1

new row3 = old row3 - 3*row1

new row3 = old row3 - 5&row2

new row3 = old row3 divided by 32

new row2 = old row2 - 8*row3

new row1 = old row1 + row3

new row1 = old row1 + row 2

Firstly is my working correct, and secondly I would think they a linearly independent for the fact that there is only one 1 in each column am I correct in thinking this?