Perhaps explicitly thinking of them as systems of equations (or transformations on a vector) would be helpful.

As an example, suppose you are asked to multiply matrices A and B, where A is [1 2, 0 4, −1 2] (the commas represent the end of a row) and B is [2 1 0, 3 1 2]. Start out by taking the rightmost matrix (B in this case) and converting it into a series of equations, one for each row. So since the first row is 2 1 0, the relevant equation will be 2x + 1y + 0z. Assign each of these equations to some other variable. So we now have

X = 2x + y

Y = 3x + y + 2z

Now do the same thing with the matrix on the left, except this time use the new variables you’ve introduced (X and Y), so the three equations you end up with (one for each row) will be

X + 2Y

4Y

-X + 2Y

Now that you have these formulae, substitute in the values of X and Y based on your earlier equations. You get

(2x + y) + 2(3x + y + 2z)

4(3x + y + 2z)

-(2x + y) + 2(3x + y + 2z)

Simplifying, you get

8x + 3y + 4z

12x + 4y + 8z

4x + y + 4z

The coefficients of these equations are the result of the multiplication. So the product of the two matrices is [8 3 4, 12 4 8, 4 1 4].

I’ll admit this is not the quickest way to go about multiplying matrices, but it might be easier for you to remember since it doesn’t seem as arbitrary. And maybe once you get used to thinking about multiplication this way, the usual visual rule will start making more sense to you.

At least some of the arguments offered by Richard Rorty in

Philosophy and the Mirror of Natureare great. Understanding the arguments takes time because they are specific criticisms of a long tradition of philosophy. A neophyte might respond to his arguments by saying “Well, the position he’s attacking sounds ridiculous anyway, so I don’t see why I should care about his criticisms.” To really appreciate and understand the argument, the reader needs to have sense of why prior philosophers were driven to these seemingly ridiculous positions in the first place, and how their commitment to those positions stems from commitment to other very common-sensical positions (like the correspondence theory of truth). Only then can you appreciate how Rorty’s arguments are really an attack on those common-sensical positions rather than some outre philosophical ideas.