Let $A_R$ be the finitely generated abelian group, determined by the relation-matrix $$R := \begin{bmatrix} -6 & 111 & -36 & 6\\ 5 & -672 & 210 & 74\\ 0 & -255 & 81 & 24\\ -7 & 255 &-81 & -10 \end{bmatrix}$$ Reduce this matrix using Smith Normal Form and determine the ...

Consider the integral matrix $$R = \left(\begin{matrix} 2 & 4 & 6 & -8 \\ 1 & 3 & 2 & -1 \\ 1 & 1 & 4 & -1 \\ 1 & 1 & 2 & 5 \end{matrix}\right).$$ Determine the structure of the abelian group given by generators and relations. $$A_r = \{a_1, a_2, a_3, a_4 | R \circ \vec{a} = 0\}...

I've been given the finitely generated abelian group: $$\langle x_1, x_2 \mid 6x_1-6x_2, -6x_1-12x_2, 4x_1-8x_2\rangle$$ and written the corresponding matrix: $$A=\begin{pmatrix} 6 & -6 \\ -6 & -12 \\ 4 & -8 \end{pmatrix}$$ I now need to reduce this to Smith Normal form using the unimodular el...

This is an homework exercise of the Algebra lecture. I need to evaluate the Smith normal form of the following matrix $$A:=\begin{pmatrix}1 & -\xi & \xi-1\\2 \xi&8&8\xi+7\\\xi& 4 & 3\xi +2 \end{pmatrix} \in M(3\times 3;\Bbb{Z}[\xi]),$$ where $\xi := \frac{1+\sqrt{-19}}{2}$. We have ...

I would like to put this matrix below into Smith Normal Form over $\mathbb{Q}[x]: $ $$\left( \begin{array}{ccc} 7 & x & 0 & -x \\ 0 & x-3 & 0 & 3\\ 0 & 0 & x-4 & 0 \\ x-6 & -1 & 0 & x+1 \end{array} \right)$$ but I am stuck here: $$\left( \begin{array}{ccc} 7 & 0 & 0 & -x \\ 0 & x & 0 & 3\\ 0 & 0