Calculation and Uniqueness of Smith Normal Forms

[jj4]

FYI this is a homework problem which I already have the answer to but would like to clarify some points on.

Homework Statement

Find the Smith Normal Form of the matrix

<br /> \left[ \begin{array}{cccc} 6 &amp; 0 &amp; 4 \\ 0 &amp; 6 &amp; 8 \\ 0 &amp; 3 &amp; 0 \end{array} \right]<br />

over the ring of integers.

Homework Equations

As I understand it Smith Normal Forms are unique up to multiplication by a unit in R.

The Attempt at a Solution

I have performed a series of elementary row and column operations to get to the following Smith Normal Form:

<br /> \left[ \begin{array}{cccc} 6 &amp; 0 &amp; 0 \\ 0 &amp; 6 &amp; 0 \\ 0 &amp; 0 &amp; -24 \end{array} \right]<br />

However the "answer" given is actually:

<br /> \left[ \begin{array}{cccc} 1 &amp; 0 &amp; 0 \\ 0 &amp; 6 &amp; 0 \\ 0 &amp; 0 &amp; -24 \end{array} \right]<br />

The only units in the ring are 1 and -1 so the answers are not equivalent and presumably I have gone wrong somewhere. Can someone help me out here. My problem is not only related to this example, I have the same issue with others. Is it because my Smith Normal Form and the original matrix are not actually equivalent?

 

[jj4]

I'm just looking for a basic explanation of how to calculate Smith Normal Forms. Is this thread in the correct sub-forum?