Hopefully at this point, you're a little bit more familiar with the linear system, a matrix, how the two connect, and how to solve an augmented matrix to get values for original linear system if they exist at all. But besides understanding what a matrix is and what a linear system is, we really need to learn another super foundational concept, which is matrix algebra. Just like if we had 1 plus 1 and we added it and we got 2, if we were dealing with algebra, we would need to know the basics of algebra. How do we add them? How do we divide them? How do we multiply them? Concepts like that translate over to a linear system and therefore they translate over to a matrix. We can multiply a matrix, we can add a matrix to another matrix or subtract a matrix, pretty much just as adding a negative. We also can scale a matrix. These three are the operations that we really need to become familiar with. There's also another concept when dealing with matrices, which is the identity matrix. We're going to need to know that just like when we do algebra, if you need to multiply by 1, you're allowed to do that, assuming you do it correctly. You can also multiply by say, 3 divided by 3. Sometimes that's helpful in dealing with fractions and changing around a normal algebraic equation. Manipulating an equation, just like we do that with an identity which in normal algebra again is 1, we're going to have an identity matrix, which we can use to manipulate matrices. Let's talk about some of these operations. When we talk about adding a matrix, we need one thing to really hold true in order to be able to add two matrices together, so if we have A which is a matrix, and B which is also a matrix, if we want to add them, then we need them to be the same dimension or have the same dimensions. If A is a 3 by 2 matrix, three rows, two columns, then in order to add B to it, we also need that to have three rows and two columns. Just like adding numbers is relatively simple, adding a matrix to another matrix is also relatively simple. Let's take a look at that. If we have a 3 by 2 matrix, like my example has here, 1, 2, negative 1, 0, 1, 7. Let's call that A. We want to add it to B, which is 7, negative 2, 6, negative 6, 5, negative 2. All we do is add this entry to this entry. We do row and column based entries, we add them to each other. The first row, first column of this matrix will be added to the first row and the first column of this matrix, so 1 plus 7 will give me 8. Here, the first row, second column plus the first row, second column, so 2 plus negative 2 will give me 0. We keep doing this second row, first column, second row, first column, so negative 1 plus 6 will give me positive 5. Go on to the next one, second row, second column, 0 plus negative 6 will give me negative 6, 1 plus 5 will give me 6, and here's 7 plus negative 2 will give me 5. This is A plus B and we can call it if you want C, and we know that C is also going to be a 3 by 2. When you add two matrices, they need to be the same dimensions, you need to have the same dimensions, and the resulting matrix that you get by adding them also has the same dimensions as the original two matrices. Next time we'll talk about scale and identity, and then one more video after that, if we can't put it in the next video, at least, we'll talk about multiplying matrices, which is a little bit more complicated, but definitely doable.
