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jj4
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FYI this is a homework problem which I already have the answer to but would like to clarify some points on.
Find the Smith Normal Form of the matrix
[tex] \left[ \begin{array}{cccc} 6 & 0 & 4 \\ 0 & 6 & 8 \\ 0 & 3 & 0 \end{array} \right][/tex]
over the ring of integers.
As I understand it Smith Normal Forms are unique up to multiplication by a unit in R.
I have performed a series of elementary row and column operations to get to the following Smith Normal Form:
[tex] \left[ \begin{array}{cccc} 6 & 0 & 0 \\ 0 & 6 & 0 \\ 0 & 0 & -24 \end{array} \right][/tex]
However the "answer" given is actually:
[tex] \left[ \begin{array}{cccc} 1 & 0 & 0 \\ 0 & 6 & 0 \\ 0 & 0 & -24 \end{array} \right][/tex]
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?
Homework Statement
Find the Smith Normal Form of the matrix
[tex] \left[ \begin{array}{cccc} 6 & 0 & 4 \\ 0 & 6 & 8 \\ 0 & 3 & 0 \end{array} \right][/tex]
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:
[tex] \left[ \begin{array}{cccc} 6 & 0 & 0 \\ 0 & 6 & 0 \\ 0 & 0 & -24 \end{array} \right][/tex]
However the "answer" given is actually:
[tex] \left[ \begin{array}{cccc} 1 & 0 & 0 \\ 0 & 6 & 0 \\ 0 & 0 & -24 \end{array} \right][/tex]
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?