I think the parallelepiped has some significance not mentioned in the usual presentation: it is the figure resulting from the matrix’s linear transformation applied to the unit axis-aligned cube between (0,0,0) and (1,1,1) in an orthonormal basis — or, in other words, { *aî* + *b?* + *ck?* | *a,b,c* ? [0,1] } — which could be considered the most basic three-dimensional shape. (I assume this also generalizes to arbitrary dimensions.)

Then the cases where the transformation is not invertible = the cases where the parallelepiped is of zero volume = the cases where the determinant is zero. We can start by asking “When is a linear transformation invertible?”, find that it is so if a unit cell is not transformed to a degenerate shape with zero volume, and obtain the determinant as the formula for that volume.

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