What is the most convenient way to introduce matrices to students?
DO contextualise them. Give them some information and ask them to organise it in an easy-to-read, logical manner. Every one of them will create a table. Explain to them that matrices are used to store lots of information, and matrix operations are used when you want to do calculations on lots of information quickly.
A good example I can think of would be two fast food stores of the same chain obviously sell the same burgers. Tell them how many burgers were sold on each particular day, and ask them to find an easy way to write that information - they'll put it in a table, which you then call a matrix. Then you could ask them things like "here are some other matrices which give how much was sold in each week for a month, how much was sold in total in that month?" (matrix addition). Use your imagination.
Also, this site gives a fantastic way to introduce matrix multiplication, using the context of running an ice-cream shop (all students love ice-cream).
Being that I am a Physicist I have no problem with this approach. Mainly the introduction in Physics starts with the rotation matrix then continues with other coordinate manipulations. However in the long run this is really only an introduction to square matrices, which are not general enough for your purposes if I understand what you are wanting to do. They make valuable examples, though, once you get into the topic a ways.
-Dan