> it is not at all clear why one would multiply matrices
I disagree. Matrix multiplication is the composition of linear maps. AB is the transformation you get by applying B, then applying A to the result of that. Furthermore this provides a great intuition for why matrix multiplication is not, in general, commutative. If your professor didn't make this clear when you learned the definition of matrix multiplication, then they didn't do a great job.
>That determinants are multiplicative is really wonderful, but not at all obvious
Again, I think it is kind of obvious. The determinant of a map measures the degree to which it alters the area/volume/measure of the unit cube. So if A doubles it and B triples it, then BA should scale it by a factor of 6, since matrix multiplication is the composition of maps, and you're tripling the size of something that was already doubled.
This is why in a first course on linear algebra, it's more important to get the intuition and logical structure down than learn advanced matrix decompositions. Now your complex number example is a little less obvious. I actually think using the 2x2 matrix representation of complex numbers is the best way to introduce them because it sidesteps the part where you take roots of negative numbers, which is the most conceptually problematic part for many people and usually requires some kind of handwave.
I disagree. Matrix multiplication is the composition of linear maps. AB is the transformation you get by applying B, then applying A to the result of that. Furthermore this provides a great intuition for why matrix multiplication is not, in general, commutative. If your professor didn't make this clear when you learned the definition of matrix multiplication, then they didn't do a great job.
>That determinants are multiplicative is really wonderful, but not at all obvious
Again, I think it is kind of obvious. The determinant of a map measures the degree to which it alters the area/volume/measure of the unit cube. So if A doubles it and B triples it, then BA should scale it by a factor of 6, since matrix multiplication is the composition of maps, and you're tripling the size of something that was already doubled.
This is why in a first course on linear algebra, it's more important to get the intuition and logical structure down than learn advanced matrix decompositions. Now your complex number example is a little less obvious. I actually think using the 2x2 matrix representation of complex numbers is the best way to introduce them because it sidesteps the part where you take roots of negative numbers, which is the most conceptually problematic part for many people and usually requires some kind of handwave.