### Motivation

I teach two courses, and two sections of each; none of my students are math majors. Most have never taken a university-level math course before, and many dread having to take this one. While some of my students show some facility with quantities, it’s probably fair to say that none are exceptionally strong in math. Fewer still enjoy it - and that’s what made this week’s class on matrices that much better.

I’m teaching the basics of algebra of matrices - how to add, subtract, and multiply them. “It’s no mistake that I left out ‘divide’ from the title of this lesson,” I told them. “You have to be careful when you talk about dividing matrices.”

The textbook I’m using (the crappy one, though it’s actually not terrible) defines matrix multiplication, gives a few small examples, and stops short of any applications whose solutions are actually simplified by matrices. With a few hours to plan class, I decided on a whim to introduce Markov chains - they had the tools to understand the basics of them, and they really showcase the usefulness of defining matrix multiplication in such an unusual way. I wrote out a sample problem on the board:

Every year, 97% of city-dwellers remain in the city, while the other 3% move to the suburbs. 94% of suburb-dwellers stay in the suburbs, and the other 6% move to the city. If there are 10000 suburbanites and 5000 city-dwellers in 2004, how many are there in 2006? In 2007? In 2008?We worked it out - I more than them, to be fair - and got a 2x2 matrix, A, representing the movement between the city and the suburbs, and a 2x1 matrix, B, giving the populations for each.

So, to get the data for 2005, I showed them, you multiply A by B. How about the 2006 data?

After some thinking, one student figured it out - multiply the matrix containing the 2005 data by A. “So,” I said, “A^{2}B. And for 2006?”

“A^{3}B,” replied one student.

At this, a student in the front row asked, “What if we wanted to know how many people were in the cities and suburbs in 2003? Can we do that?”

Before I could answer, a classmate tilted her head and mused, “Well, we know where people are this year, and we know how they move around. So we should be able to figure it out.”

And suddenly half a dozen students were trying:

“Yeah - well, ok, you have AB, and A^{2}B, and A^{3}B to find the information in the future -”

“…and *no* A’s to find this year - like A^{0}B -”

“Yeah, so you can just divide by A - take A^{-1}B! And A^{-1}, that’s like one *over* A”.

At this I said, “One over A - what does that mean?”

They took over again:

“Well, you divide - ”

“But remember, at the beginning of class, she said you *can’t divide* - ”

“No, she said you have to be *careful* - but why?”

“How,” I interrupted, “Would you go about dividing 1 by A?”

There was some thought: “Well, maybe the first element is one over the first element of A, and -”

“No, it can’t be that, multiplication with matrices is weird, so division should be weird too…”

“And why should you be able to divide a *number* by a matrix? You can’t *add* a number to a matrix -”

“But you can *multiply* a matrix by a number -”

I interrupted again: “How would you check your answer, if you have a matrix 1/A?”

Some thought, and then a voice from the back - “Well, when you mutiply *that* by A, you get back one - wait - you can’t get a number when you multiply matrices, you get another matrix.”

There was a long pause. The second hand of the clock was approaching the half-hour, and a lot of my students had another class after mine.

“Okay, so tell us, how *do* you divide by a matrix?”

“*Can* you divide by a matrix? Is that what we’re supposed to do to get the information for 2003?”

I grinned. “I’ll see you on Friday,” I said.