Let a,b,c be any real number. Consider the following 3 x 3 matrix: 

What kind of effect does this matrix have in 3D space? Consider an arbitrary coordinate [x,y,z], then transform this point by the above matrix to get:

So this type of matrix only effects the x,y direction; furthermore the effect is proportional to y and z leading to quick changes as a,b,c are increased in magnitude. This stretching may be extreme but remember the determinant of the above is always 1 so the inverse is well-posed. Consider the unit sphere in 3D pictured below, it was then transformed using small values for a,b,c:

It turns out that the set of all such matrices forms a mathematical group. To see this, consider another such member for different choices of real numbers, alpha, beta,and gamma.  

The matrix product AB of course is not necessarily the same as BA given the non-communativity of matrix multiplication in general. But is there a condition on the real numbers a,b,c,alpha,gamma,beta so that they do indeed commute? After a few intimate minutes with scratch paper one arrives at: 

The products do end up being very similar, except for the 1st row 3rd column entry. We then require the following equality hold: 

Notice this is the same as saying that the ratio b/a and beta/alpha are proportional, presupposing that alpha and a are nonzero. 

Another interesting result is the inverse of A (the undoing of the sphere deformation) is easily calculated as: 

The code for producing this linear transformation as well as its inverse is downloadable here.