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1
There is a bit of theory to go through to explain how the invariant factors of the matrix you wrote down is actually related to $M,$ you will have to go through your notes to see the precise connection.
As for the calculation, perhaps recheck your operations, or post them here so we might we able to say where you went wrong.
It turns out that for this ...
1
When you do the operation $4 R_2 - 5 R_1$, you are altering the determinant by a factor of $4$, since you are multiplying your matrix on the left by
$$
\begin{bmatrix}1 & 0 & 0\\-5&4&0\\0&0&1\end{bmatrix}.
$$
Ditto for $3 R_1 + R_2$, which alters the determinant by a factor of $3$. These two operations have introduced the spurious ...
1
$\operatorname{im}(\varphi)=\{\varphi(x_1,x_2,x_3):(x_1,x_2,x_3)\in\mathbb Z^3\}=\{x_1\varphi(1,0,0)+x_2\varphi(0,1,0)+x_3\varphi(0,0,1):$ $(x_1,x_2,x_3)\in\mathbb Z^3\}=\{x_1(6,4)+x_2(4,8)+x_3(4,0):(x_1,x_2,x_3)\in\mathbb Z^3\}$ $=$ $\langle(6,4),(4,8),(4,0)\rangle=N$.
The form of the canonical diagonal matrix $D$ shows that there exists a $\mathbb ...
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