My Linear Algebra text has some introductory examples of interpreting the determinant as a scaling factor. These all involve interpreting the columns of a matrix B as coordinates to form a square or triangle, then showing that when left multiplying B by another matrix A that has a determinant, det(A), that the area of the shape formed by interpreting the columns of the resulting matrix as coordinates is scaled by det(A). That part all makes sense.
Later they introduce this proposition:
The first confusing thing is they're saying a 3x3 matrix is a box -- that's not true if you interpret the columns as coordinates like before; you need more columns. At best 3 points could describe a plane or a triangle in a plane. If I interpret their A matrix to be a matrix that we multiply by to scale another matrix B with columns as coordinates, I can see why (kA)B=C_1 would have factors that are k times longer than those in just AB=C_2, because the rows of A each in essence describe a linear combination for how to produce the new columns of C_1 and C_2, and if that sum is multiplied by k every column vector in C_2 should be k times longer than in C_1. Then if we separately already accept that det(A) is the scaling factor for the area of a new shape produced when left multiplying by A, I can see the proposition. But in that line of reasoning kA never represents a box; you could interpret it to be one in the sense that you can interpret 3 vectors with a length as describing a cube, but that's inconsistent with how they interpreted things before and it's not clear why that interpretation is supposed to help.
Am I interpreting this correctly? Or is there an easier way of looking at it?