find cokernel of a matrix
Solution 1:
All of your work is correct, well done.
Regarding the Smith normal form: we have $UTV = S$, with the $S$ that you wrote and $$ U = \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 2 & 0 & 0 & -1 \\ -35 & -1 & 0 & 20 \\ -1 & 0 & 1 & 0 \\ \end{array} \right), \quad V = \left( \begin{array}{ccc} 1 & -4 & -5 \\ 0 & -1 & -2 \\ 0 & 2 & 3 \\ \end{array} \right). $$