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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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    [Smith Normal form](http://en.wikipedia.org/wiki/Smith_normal_form)?2011-12-15
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    Of your interest: $v_k$ is equal to the GCD of all $k \times k$ minors of $A.$ And $v_k$ divides of coefficient of $x^k$ in the characteristic polynomial of $A.$2011-12-15
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    I've just noticed the ring-theory tag. Take a look at *[Linear Algebra Over Commutative Rings* by Bernard R. McDonald](http://books.google.ca/books/about/Linear_algebra_over_commutative_rings.html?id=hkCgw_5wRq4C&redir_esc=y). There is little about Smith Forms, though (pp 63. in the 1984 edition) and in the exercises IIRC.2011-12-15
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    Thank you for your help. I don't understand your first comment, J.D. . What do you mean with minor of A ? In b) How can one show that the entries of $P'^{-1}, X^{-1}$ are also in $\mathbf{Z}$, and what about the determinants? that is not really clear from the wiki article method, either. Help is greatly appreciated.2011-12-15
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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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