My dad figured this out years ago.
The string method work for all matrices, and it is at least ten times quicker to “do” than to “write about”.
Matrix multiplication without tears!
Filed under algebra, education, math, teaching, transformations, Uncategorized
Saving school math 
You had me at line 1. But then it took me 3 goes.
{ I am prompted to write as I recently asked my Dad (not a Math-man) what he knew about X (moderately new in an offshoot field to his area). “Never heard of it”. He had met Nelkon though. The Physics-text legend. (BTW I arrive here via ur comment praising Gelbaum). I don’t have any copies of those school books, indeed all my undergrad books have gone too. }
Anyway my point is that the 3D example you illustrate reminded me of my best lecture: Palindrome testing in a particular computer language. Twenty years later and my lecture notes (too) are gone but the new fangled way is:
http://stackoverflow.com/questions/8362572/check-if-a-string-is-palindrome
What is interesting to us Programmers is the feud over what is good and bad programming. I only remember one message (of experience) that I had to give when I was teaching the language C: “String think pointer (to char)” …… the first is a vector, the second: I would announce at the start of my lecture courses was my purpose.