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\end{bmatrix}.</math>
Note that <math>A</math> and <math>B</math> are two distinct matrices in the row echelon form, which would mean that their span is the same if they're treated as matrices over some field. Moreover, they're in the [[Hermite normal form]], meaning that their row span is also the same if they're considered over <math>\Z</math>, the [[ring of integers]].<ref name=":0" />
However, <math>\Z_{12}</math> is not a field and over general rings it is sometimes possible to nullify a row's pivot by multiplying the row with a scalar without nullifying the whole row. In this particular case,
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