Measure of Image of Linear Map

Hint 1) Enough to show this in the case that $A$ is an $n$-dimensional parallelopiped (as John M pointed out).

Hint 2) Recall from linear algebra that any linear mapping can be written as a composition of elementary linear mappings of three types: (usually expressed in the language of matrices, so I will do the same here) A) swap two rows, B) multiply a row by a scalar, C) add a scalar multiple of one row to another.

Hint 3) Swapping two coordinates is geometrically a reflection with respect to a hyperplane, so type A is easy. Type B amounts to stretching one of the coordinates. Type C is geometrically a shearing, i.e. the type of mapping that turns a rectangle into a parallelogram with same base and height.

I've taken a quick look at the proof given in the text that you reference. It largely follows Jyrki's approach, but with a small difference. The text (in part (v) of the proof) considers these type C shearing matrices, but with only a multiple of one, rather than a general multiple, and then refers to a rather specific theorem that allows for decomposition into elementary matrices, such that the elementary matrix with addition of one row to the next only requires a multiple of one. This theorem is stronger than the usual decomposition theorem, and I haven't been able to find a convenient reference for it.

Anyway, Jyrki's proof is nicer than your text's: There is no reason to restrict the multiple of your shearing matrix to one - the same argument goes through for any multiple. Once you allow for this general shearing matrix, you can then refer to the more standard proofs of decomposition into elementary matrices. I like Ch 1 of Artin's "Algebra" for this.

For another approach which might be quite illuminating, see Ch 5 of Lax's "Linear Algebra". He starts with the properties of what an operator for "signed volume" must look like, and then he deduces a formula which turns out to be the usual determinant.