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

$\displaystyle

\left[ {\begin{array}{*{20}c}

1 & { - 1} & { - 1} \\

{ - 2} & 3 & {10} \\

3 & 2 & 5 \\

\end{array}} \right]

$

now start gauss-jordon elimination.

new row2 = old row2 + 2*row1

$\displaystyle

\left[ {\begin{array}{*{20}c}

1 & { - 1} & { - 1} \\

0 & 1 & 8 \\

3 & 2 & 5 \\

\end{array}} \right]

$

new row3 = old row3 - 3*row1

$\displaystyle

\left[ {\begin{array}{*{20}c}

1 & { - 1} & { - 1} \\

0 & 1 & 8 \\

0 & 5 & 8 \\

\end{array}} \right]

$

new row3 = old row3 - 5&row2

$\displaystyle

\left[ {\begin{array}{*{20}c}

1 & { - 1} & { - 1} \\

0 & 1 & 8 \\

0 & 0 & { - 32} \\

\end{array}} \right]

$

new row3 = old row3 divided by 32

$\displaystyle

\left[ {\begin{array}{*{20}c}

1 & { - 1} & { - 1} \\

0 & 1 & 8 \\

0 & 0 & 1 \\

\end{array}} \right]

$

new row2 = old row2 - 8*row3

$\displaystyle

\left[ {\begin{array}{*{20}c}

1 & { - 1} & { - 1} \\

0 & 1 & 0 \\

0 & 0 & 1 \\

\end{array}} \right]

$

new row1 = old row1 + row3

$\displaystyle

\left[ {\begin{array}{*{20}c}

1 & { - 1} & 0 \\

0 & 1 & 0 \\

0 & 0 & 1 \\

\end{array}} \right]

$

new row1 = old row1 + row 2

$\displaystyle

\left[ {\begin{array}{*{20}c}

1 & 0 & 0 \\

0 & 1 & 0 \\

0 & 0 & 1 \\

\end{array}} \right]

$

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?