In this video we'll talk about matrix addition and subtraction, as well as how to multiply a matrix by a number, also called Scalar Multiplication. Let's start an example. Given two matrices like these, let's say I want to add them together. How do I do that? And so, what does addition of matrices mean? It turns out that if you want to add two matrices, what you do is you just add up the elements of these matrices one at a time. So, my result of adding two matrices is going to be itself another matrix and the first element again just by taking one and four and multiplying them and adding them together, so I get five. The second element I get by taking two and two and adding them, so I get four; three plus three plus zero is three, and so on. I'm going to stop changing colors, I guess. And, on the right is open five, ten and two. And it turns out you can add only two matrices that are of the same dimensions. So this example is a three by two matrix, because this has 3 rows and 2 columns, so it's 3 by 2. This is also a 3 by 2 matrix, and the result of adding these two matrices is a 3 by 2 matrix again. So you can only add matrices of the same dimension, and the result will be another matrix that's of the same dimension as the ones you just added. Where as in contrast, if you were to take these two matrices, so this one is a 3 by 2 matrix, okay, 3 rows, 2 columns. This here is a 2 by 2 matrix. And because these two matrices are not of the same dimension, you know, this is an error, so you cannot add these two matrices and, you know, their sum is not well-defined. So that's matrix addition. Next, let's talk about multiplying matrices by a scalar number. And the scalar is just a, maybe a overly fancy term for, you know, a number or a real number. Alright, this means real number. So let's take the number 3 and multiply it by this matrix. And if you do that, the result is pretty much what you'll expect. You just take your elements of the matrix and multiply them by 3, one at a time. So, you know, one times three is three. What, two times three is six, 3 times 3 is 9, and let's see, I'm going to stop changing colors again. Zero times 3 is zero. Three times 5 is 15, and 3 times 1 is three. And so this matrix is the result of multiplying that matrix on the left by 3. And you notice, again, this is a 3 by 2 matrix and the result is a matrix of the same dimension. This is a 3 by 2, both of these are 3 by 2 dimensional matrices. And by the way, you can write multiplication, you know, either way. So, I have three times this matrix. I could also have written this matrix and 0, 2, 5, 3, 1, right. I just copied this matrix over to the right. I can also take this matrix and multiply this by three. So whether it's you know, 3 times the matrix or the matrix times three is the same thing and this thing here in the middle is the result. You can also take a matrix and divide it by a number. So, turns out taking this matrix and dividing it by four, this is actually the same as taking the number one quarter, and multiplying it by this matrix. 4, 0, 6, 3 and so, you can figure the answer, the result of this product is, one quarter times four is one, one quarter times zero is zero. One quarter times six is, what, three halves, about six over four is three halves, and one quarter times three is three quarters. And so that's the results of computing this matrix divided by four. Vectors give you the result. Finally, for a slightly more complicated example, you can also take these operations and combine them together. So in this calculation, I have three times a vector plus a vector minus another vector divided by three. So just make sure we know where these are, right. This multiplication. This is an example of scalar multiplication because I am taking three and multiplying it. And this is, you know, another scalar multiplication. Or more like scalar division, I guess. It really just means one zero times this. And so if we evaluate these two operations first, then what we get is this thing is equal to, let's see, so three times that vector is three, twelve, six, plus my vector in the middle which is a 005 minus one, zero, two-thirds, right? And again, just to make sure we understand what is going on here, this plus symbol, that is matrix addition, right? I really, since these are vectors, remember, vectors are special cases of matrices, right? This, you can also call this vector addition This minus sign here, this is again a matrix subtraction, but because this is an n by 1, really a three by one matrix, that this is actually a vector, so this is also vector, this column. We call this matrix a vector subtraction, as well. OK? And finally to wrap this up. This therefore gives me a vector, whose first element is going to be 3+0-1, so that's 3-1, which is 2. The second element is 12+0-0, which is 12. And the third element of this is, what, 6+5-(2/3), which is 11-(2/3), so that's 10 and one-third and see, you close this square bracket. And so this gives me a 3 by 1 matrix, which is also just called a 3 dimensional vector, which is the outcome of this calculation over here. So that's how you add and subtract matrices and vectors and multiply them by scalars or by row numbers. So far I have only talked about how to multiply matrices and vectors by scalars, by row numbers. In the next video we will talk about a much more interesting step, of taking 2 matrices and multiplying 2 matrices together.