Simply use row reduction: the rank is the number of non-zero rows after you've performed row reduction:

\(\begin{bmatrix}3 & 2 & -1\\2 & -3 & -5\\-1 &-4 &-3 \end{bmatrix}\rightsquigarrow\begin{bmatrix}1 & 4 & 3\\3 & 2 & -1\\2 &-3 &-5 \end{bmatrix}\rightsquigarrow\begin{bmatrix}1 & 4 & 3\\0 & -10 & -10\\0 &-11 &-11 \end{bmatrix}\rightarrow\begin{bmatrix}1 & 4 & 3\\0 & 1 & 1\\0 &-11 &-11 \end{bmatrix}\rightsquigarrow\begin{bmatrix}1 & 4 & 3\\0 & 1 & 1\\ 0 &0 &0 \end{bmatrix}\)