College Algebra: An Introduction to Matrices

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Now, it's one thing to look at a number, say 5. But the kinds of things that we've been looking at recently have been actually collections of equations. In fact, what we've looked at for a while was we were actually looking at systems of linear equations, where you have a lot of equations, and you want to solve them at once. And you can have use either the substitution method or an elimination method or, and so forth. The idea is there are a lot of numbers. It's not just 5 anymore. Now, you've got a ton of numbers that you've got to keep track of.
If you think about it, when you look at some sort of system like this one, 2x - 3y = 1. 4x + 7y = 5. Really, if you think about it, the real key players here, are the co-efficients, the 2, the -3, the 1, the 4, the 7, and the 5. In fact, the x's and the y's aren't even important as long as you sort of remember where they belong. For example, if you take this and this and flop them, if you promise to keep the x's always here and always the y's here, you can perform all the techniques we talked about of elimination by just manipulating these numbers.
Really, all that matters, in fact, are the numbers themselves. As long as we keep things on the up and up, we could just keep those numbers in the way they are and display them all adjacent, juxtaposition if you will, somehow we should just be able to manipulate that. Then at the end of the day say, whatever is in here, that's the x's, these are the y's, and these are the answers. So somehow just manipulating these numbers, we might be able to make some progress in solving these types of equations.
Instead of just thinking about 5 now, I want you to think about a whole list of numbers, a whole collection of numbers at once that form a block. These objects actually are called matrices. A matrix is just a block of numbers. It's a way of expressing a whole bunch of numbers at once, rather than just 5.
So, let me just introduce you to some of the basic forms of matrices, and how they appear, and how we look at them, and so forth. This is an example of a matrix. You can see that it's just a block of numbers. This is [1, 3, 2, 1, 4, 0,] and so forth.
First of all, how would you read this? I would say that this is a 2 by 3 matrix, and the reason is because there are two rows and three columns. These things here are columns, and these here are rows. When I talk about the size of a matrix, I'm talking about how many rows, and how many columns. This would be a 2 by 3, two rows and three columns.
Here's another matrix, different size. This would be actually a 1 by 3, because one row and three columns. Here is the row, and these are the columns. This is 1 by 3. What about this? This would be 3 by 3, so this is size 3 by 3. There are nine numbers in this one. This is 3 by 2, and it looks rectangular. It's 3 by 2. This is 3 by 1. One, two, three by just one column. Here we have two rows and three columns, so it's a 2 by 3. We always give this measurement first, how many of these, and then how many of these. Here we have 3 by 4, because we have four columns and three rows, so say it's a 3 by 4.
Anyway, that's the way we denote them. That's how they look. Now, what we're going to see is how they actually have some power by just manipulating, not just 5, but now, we're going to manipulate all of these numbers at the same time.
We're going to start to do algebra and arithmetic on collections of numbers, or on an array of numbers, and these are all known as matrices. We'll take a look at how all these go up next.
Systems of Equations
Matrices
An Introduction to Matrices Page [1 of 1]