Matrix Multiplication, A Novella: Chapter 1

So you got hired as an event planner. Congratulations on your new job!

You’re a clever event planner, and therefore you know that your events need tables for people to eat at. You rent tables per-event from your favorite party supply place, Vendor 1. Here’s the spreadsheet you use to track costs:

(I hope you didn’t just ignore that spreadsheet. If your eyes glazed over, go back and read it. It’s simple.)

Everything looks great! However, two nights before Gala A, you wake up in a cold sweat: you forgot chairs. You figure out how many chairs each event needs, call Vendor 1 to get a price, and re-tool your spreadsheet. Nice save! Now your spreadsheet looks like this (click for bigger, and don’t ignore this one either):

It’s one night before Gala A, and your phone is ringing at 2AM. What could it be? It’s Vendor 1’s archrival, Vendor 2! She wants to scoop her old boss, and she has a deal for you: chairs cost $7 each, but tables are only $8 each. You can’t mix and match tables from Vendor 1 with chairs from Vendor 2 — ever since Vendor 2 keyed Vendor 1’s car, they won’t work together. (Click for bigger)

Looks like Vendor 2 is cheaper for Gala A, Soiree B, and Reception C. You should keep Vendor 1 for Banquet D, though!

Man, you’re good at this event planning stuff. Two months pass, and you’re an old pro. This is what your spreadsheet looks like now:

More events! vendors! More items to rent! This is intense. Frankly, now that you’re a big-shot, these labels are looking like overhead. So you clean it up:

These are the same spreadsheets, just with all the non-math information chopped out. You decide to call these math-only spreadsheets matrices. To keep things from getting confusing, you implement some rules:

  • Matrix A holds item-to-rent details per event. A row represents a single event, and a column represents the number of a single item-type to rent for each event.
  • Matrix B holds vendor prices per item-to-rent. A row represents a single item-type, and a column represents a vendor’s prices for each item.
  • If we want to add a new item-type (for instance, lamps) then we need to know how much each vendor charges per-lamp. This means that the number of columns in A (representing item-types) must match the number of rows in B (since they also represent item-types).

Now, you’ve removed everything but the sweet, sweet math. And once you do that, something surprisingly simple shows up:

That is… (click for bigger)

And you just invented matrix multiplication. Congratulations!

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