# Homework Help: Calculation and Uniqueness of Smith Normal Forms

1. Jan 13, 2012

### jj4

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

1. The problem statement, all variables and given/known data

Find the Smith Normal Form of the matrix

$$\left[ \begin{array}{cccc} 6 & 0 & 4 \\ 0 & 6 & 8 \\ 0 & 3 & 0 \end{array} \right]$$

over the ring of integers.

2. Relevant equations

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

3. The attempt at a solution

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

$$\left[ \begin{array}{cccc} 6 & 0 & 0 \\ 0 & 6 & 0 \\ 0 & 0 & -24 \end{array} \right]$$

However the "answer" given is actually:

$$\left[ \begin{array}{cccc} 1 & 0 & 0 \\ 0 & 6 & 0 \\ 0 & 0 & -24 \end{array} \right]$$

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?

2. Jan 14, 2012

### jj4

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