Take the following:

where M is a matrix and b1 and b2 are the basis vectors for the coordinate frame. b1 and b2 are defined as:

so our matrix looks like:

i.e. column major matrix.

The determinant of a scalar is simply the scalar itself. This makes sense as the 'area' of a scalar would be itself (please correct me if I sound delusional) - just using the concept of area applied to a scalar to provide a concept for the determinant, which ultimately returns an area right?

The determinant of a 2x2 matrix M is:

abs(|M|) is equal to the area of the parallelogram bounded by the basis vectors b1 and b2.

I'm struggling to see the relationship of the determinant formula, i.e. how does the x component of the first basis vector multiplied by the y component of the second basis vector have significance?

Currently I'm grasping the idea that the determinant forumula is done this way to compute the signed area of the parallelogram, thus 'preserving' the 'handedness' of the coordinate frame, but beyond that I'm lost...

I tried computing the area of the parallelogram bounded by the two basis vectors, as:

where ||x|| is the magnitude of vector x.

This should work fine, because the area of a parallelogram is the product of its adjacent sides. However I can't find anything relating the determinant to this calculation.