I am struggling to calculate the Smith Normal Form of this matrix. I know it is wrong because I checked on a computer. Can someone help me where I am going wrong?
$\begin{bmatrix} 6&2&3&0\\ 2&3&-4&1\\ -3&3&1&2\\ -1&2&-3&5 \end{bmatrix}\implies \begin{bmatrix} 0&6&2&3\\ 1&2&3&-4\\ 2&-3&3&1\\ 5&-1&2&-3 \end{bmatrix}\implies \begin{bmatrix} 1&2&3&-4\\ 0&6&2&3\\ 2&-3&3&1\\ 5&-1&2&-3 \end{bmatrix}\implies \begin{bmatrix} 1&0&0&0\\ 0&6&2&3\\ 2&-7&-3&9\\ 5&-11&-13&17 \end{bmatrix}\implies \begin{bmatrix} 1&0&0&0\\ 0&6&2&3\\ 0&-7&-3&9\\ 0&-11&-13&17 \end{bmatrix}\implies \begin{bmatrix} 1&0&0&0\\ 0&2&3&6\\ 0&-3&9&-7\\ 0&-13&17&-11 \end{bmatrix}\implies \begin{bmatrix} 1&0&0&0\\ 0&2&3&0\\ 0&-3&9&2\\ 0&-13&17&28 \end{bmatrix}\implies \begin{bmatrix} 1&0&0&0\\ 0&2&1&0\\ 0&-3&12&2\\ 0&-13&30&-50 \end{bmatrix}\implies \begin{bmatrix} 1&0&0&0\\ 0&1&0&0\\ 0&0&-27&2\\ 0&0&-73&-50 \end{bmatrix}\implies \begin{bmatrix} 1&0&0&0\\ 0&1&0&0\\ 0&0&2&-27\\ 0&0&-50&-73 \end{bmatrix}\implies \begin{bmatrix} 1&0&0&0\\ 0&1&0&0\\ 0&0&2&-1\\ 0&0&-50&-723 \end{bmatrix}\implies \begin{bmatrix} 1&0&0&0\\ 0&1&0&0\\ 0&0&-1&0\\ 0&0&-645&-1396 \end{bmatrix}\implies \begin{bmatrix} 1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1396 \end{bmatrix}$
However the computer says the answer is supposed to be $$\begin{bmatrix} 1&0&0&0\\ 0&1&1&0\\ 0&0&1&0\\ 0&0&0&610 \end{bmatrix}$$
What am I doing wrong?