Is this approach to proving the multilinearity of the determinant valid (using Smith normal form)?

I'm working on an exercise in linear algebra involving the multilinearity of the determinant. The task is:

Let $𝐾$ be a field. Consider the determinant as a function:

\begin{align} &\det : K^n \times \dots \times K^n \to K\,, \\ &\quad (s_1, \dots, s_n) \mapsto \det(s_1, \dots, s_n) ] \end{align}

Show that this map is multilinear, i.e., linear in each component. (As you can see in the added picture)

the exercise

This was an optional problem, but I decided to try it for practice.

I worked on it myself first. You can see my handwritten attempt here:

my attempt

Later, I checked other solutions and saw that most of them use the Leibniz formula for determinants. I tried a different path using the Smith normal form instead.

Question:

Is this an acceptable method for proving multilinearity? Or is it considered invalid for some reason? I’m aware that I didn’t show full linearity yet (e.g. f(a⋅v)=a⋅f(v)), but I’m mostly interested in whether my overall idea/approach makes sense.

Thanks a lot in advance — I’m still new to this, so I appreciate any feedback!

Also if there is anything unclear or not appropriate about my post, I’d really appreciate it if someone could leave a comment or brief explanation before voting to close or delete it. I’m happy to edit and improve the question — but I can only do that if I understand what’s wrong. Thanks a lot!