Z-Smith Normal Form

Question

I am a bit confused on how to put the matrix \begin{bmatrix}4&2&4\\3&3&4\\2&2&2\end{bmatrix} in $\mathbb{Z}$-Smith Normal Form.

I know this can be done using unimodular elementary row and column operations, but everytime i've tried I end up with the wrong answer.

Any help will be very much appreciated

Answer 1

I perform a series of row and column operations to get $$\begin{pmatrix} 2&0&0\\ 0&2&0\\ 0&0&1 \end{pmatrix}$$

Is this your right answer ?

Answer 2

The Smith normal form of that matrix is $$\pmatrix{1&0&0\\0&2&0\\0&0&2}.$$ That the top left value is going to be $1$ is pretty obvious; just subtract the second row of the original matrix from the first.