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Suppose $m,n \in \mathbf{N}, m\le n$. Let $A$ be a matrix with $\mathbf{Q}$ linearly independent $b_{1},...,b_{m}$ in $\mathbf{Z}^{n}$.

a) Show that there are $v_{1},...,v_{m} \in \mathbf{N}$ so that $v_{1}| ... v|v_{m}$ and square matrices P', X with entries in $\mathbf{Z}$ so that A=P'TX , and T has the entries $v_{i}\delta_{ij}; i=1,...,m ; j= 1,...,n$

b) Show that P' and $X$ are invertible and that the entries of P'^{-1} and $X^{-1}$ are also in $\mathbf{Z}$

c) Show that det P' = \pm 1 and $det X = \pm 1$ and that the gcd of all entries of A is equal to $v_{1}$

This is a question from an old mock examination (which consists only of this one problem... huh).

As written in the comments, the construction in wikipedia in the smith normal form article doesn't seem to disclose much information for c) and the entries of the invertible matrices being in $\mathbf{Z}$ in b). Help is greatly appreciated.

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    @PumaDAce: All row-reduction and column-reduction operations are invertible over $\mathbb{Z}$, because all we do is exchange rows/columns and add (integer) multiples of one row/column to another row/column. So $P$ and $X$ are products of elementary matrices, each of which has integer coefficients and determinant $\pm 1$. It follows that $P$ and $X$ have integer coefficients and determinant $\pm 1$, and so their inverses (which can be computed using the cofactor matrix and dividing by the determinant) must also have integer coefficients.2011-12-16

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