You can imagine the determinant as the volume of the object spanned by its column vectors. If they are linearly independent, then they span a parallelepiped with positive volume. If not, then they span a hyperplane. But a lower dimensional object in a higher dimensional space has no volume.
The algebraic method is faster, but delivers no insights. If ##A## is invertible, then this
means there is a matrix ##B## such that ##A\cdot B = 1##. As the determinant respects multiplication we get from this ##\det(AB)=\det(A)\det(B)=\det(1)=1## and so that ##\det(A)## is a divisor of ##1##, i.e. especially not equal ##0##.
If you have only a formula for the determinant, then you should prove this homomorphism property first, e.g. per induction.
This is why I asked you, what determinant means to you.
"A is linearly independent" is not enough. What does that mean? If I take what you wrote, then I see a vector ##A## in the vector space of linear functions on ##\mathbb{R}^n##. As a single vector is always linear independent as soon as it is different from the zero vector, this statement is trivially true and has absolutely nothing to do with the determinant of ##A##. Hence I assumed, that you meant something else. However, the
closest possibility to interpret what you might have meant, is to translate it by "the column vectors of ##A## are linear independent in ##\mathbb{R}^n##. But the I have ##n## linear independent vectors, and of course they span an ##n-##dimensional space. So where is the problem?
This is why I asked you about the meaning of "A is linearly independent". Linear independence
always requires a reference: Where do the vectors live? When did you switch from matrix to vector? Matrices can only be linear independent if considered as vectors in some vector space.
The rank formula is the best way to see the implication from ##A## to its column vectors.
The remaining implication ##\det(A)\neq 0 \Longrightarrow ## "column vectors of ##A## are linearly independent" is best seen the other way around. Assume a non trivial linear dependence of the column vectors and see what is does to the determinant. One can handle this with the properties of the determinant, or the properties of ##A## as linear mapping.
This is why I asked you what ##A## means to you: a number scheme or a linear function?
Here is a good read about the geometry of such things:
https://arxiv.org/pdf/1205.5935.pdf
I don't know whether it will answer your questions above, but it will definitely help you to imaginate the objects you are dealing with.