Smith Normal Form of this polynomial non square matrix in $\mathbb Q[t]$

Question

Given $$A=\begin{bmatrix}t^3-t^2&t^2-t\\2t^3&t^4-t\\t&2t^3-2t^2\end{bmatrix} \in M_{3\times 2}(\mathbb Q[t])$$ I'm looking for the matrices $S,P,Q$ such that $S=QAP$, where $Q$ are the row and $P$ are the column transformations and $S$ is its SNF.

I have tried using the euclidean algorithm, but I can't reach SNF, since there happen to be ridiculously high fractions. After that step I struggle to find any reasonable answer:

$$ \begin{bmatrix}t&0\\0&-4t^5+5t^4-t\\0&-2t^5+4t^4-2t^3+t^2-t\end{bmatrix}$$

Any help is appreciated.

Answer 1

Hint:

You have to find a Bézout's relation between $f=-4t^5+5t^4-t$ and $g=-2t^5+4t^4-2t^3+t^2-t$, with the extended Euclidean algorithm. You should find $$uf+vg=t(t-1).$$ Pre-multiplying by the $3\times3$ matrix: $$\begin{bmatrix}1&0&0\\0&u&v\\0&\dfrac g{t(t-1)}&\dfrac f{t(t-1)}\end{bmatrix}$$ to make arithmetic operations on rows yields the Smith normal form $$\begin{bmatrix}t&0\\0&t(t-1)\\0&0\end{bmatrix}$$

Answer 2

Since $-4t^5+5t^4-t=-t(t-1)(4t^3-t^2-t-1)$ and $-2t^5+4t^4-2t^3+t^2-t=-t(t-1)(2t^3-2t^2-1)$, you only need to find an integer multiple of the GCD of $4t^3-t^2-t-1$ and $2t^3-2t^2-1$ using Euclidean algorithm. This should be straightforward: $$ \begin{align} 4t^3-t^2-t-1 &= 2(2t^3-2t^2-1) + (3t^2-t+1)\\ 9(2t^3-2t^2-1) &= (6t-4)(3t^2-t+1) - 5(2t+1)\\ 4(3t^2-t+1) &= (6t-5)(2t+1) + 9. \end{align} $$