 Our next topic is matrix arithmetic, and again the idea to think about here is that whenever a mathematician is introduced to a new object, the first question he or she asks is generally going to be, okay, what can I do with this? How can I apply some concept that I have used in some context for some type of object? How can I apply that to this new type of object? And so we know how to perform arithmetic on real numbers, and we've introduced some of the concepts of applying arithmetic to vectors, so the natural question to ask is what sort of things can we do with matrices? Well the beginning point is we have to start by identifying when some things are equal, and so we'll define two matrices A and B. We'll say they're equal if and only if Aij equals Bij. Remember the component, the i-th row, j-th component of A, and the i-th row, j-th component of B, if they are equal for all i and j. So for example, let's consider a bunch of matrices A, B, C, and D, and let's see. What do we have here? Well B and D are equal because they are component-wise equal, the 1, 1 entry, 0, the 1, 1 entry, 0, the row 1 column 2 entry 3, row 1 column 2 entry 3, and so on. So B and D are equal matrices. A and B are not equal. They have the same entries, 1, 2, 3, and 0, but they are in different locations. The 1, 1 entry of A is 1, the first row, first column entry of B is 0, and they're not equal, and we only get to declare them equal if the equality holds for all components. So A and B are different. How about A and C? Well we might wonder about that. Well A and C, well they look like that's 1, 2, 0, 3, 1, 2, 0, 3, and the only difference is that C has this extra row, extra column of 0s here, but they are different matrices, and again notice that C has different numbers of rows and columns, and what this means is that C has a 2, 3 entry. C has a second row, third column entry. It's going to be 0. A does not. A's second row, third column entry is non-existent. It is undefined, so these two matrices are not equal. AIJ has to equal BIJ for all components. Here C has a component, 1, 2, 3, A does not, 0, and undefined are not equal, so A and C are also unequal matrices. Well we, if we think about a matrix as an extension of the vector or the tuple concept, then addition and scalar multiplication, we've already defined them for vectors, so we can try and define them in the same way for matrices, and so we'll define these two things component-wise. So let A and B be 2M by N matrices. They have to be of the same size. We'll define their sum A plus B equal to C, and we'll define that component-wise. The matrix, the entries of C, are going to be the sum of the corresponding entries of A and B. And likewise for a scalar multiplication, if I have a matrix, I can define the scalar multiple of that matrix to be the matrix whose components are k times the corresponding components. And this is essentially what we do with vectors, so there really isn't that much difference between the two, except that we're dealing with sets of vectors instead of single vectors themselves. So if I have two matrices A and B, if I want to find the sum A plus B, well I'll add those matrices component-wise. So my first component of the sum will be the sum of 3 plus 1, 1 plus 2, 2 plus 1, 0 plus negative 1, and all of these are sums of real numbers, so I can add them together without any difficulty. Another useful thing that we do with matrices is called the transpose of a matrix. So given any matrix A, I can write the transpose of A, A t, this is a superscript, this should not be read as an exponent. There's only so many ways we can write things, so a lot of things will look alike, but we'll be using them in different contexts. So this is going to be the transpose of the matrix, and it's going to be some matrix B, where B ij equals A ji. Now the algebra here isn't quite obvious what we're doing, and so until we actually do it, it's not clear why we would call this the transpose. So this is still an algebraic operation, but it's helpful to actually see it and then get some sort of visual picture of what's going on. So for example, let's take the matrix A, looking something like this, and I'm going to find a transpose that's going to be a matrix B, where B is going to be, well I'll fill in the entries of B using this formula. So let's see, B11, I'm going to reverse those components, so i throw jth column becomes jth row ith column. Well they're both one here, so that reversal doesn't do anything. B11, the first end, first row, first column entry, is going to be the same as the first row, first column entry, that's going to be three, so I'll just move that over into B. B12, first row, second column, I'll reverse those coordinates you can think of them as. So this is going to be the A21 entry, the first row, second column entry is going to be the same as the second row, first column entry. So that's this two is going to go over here, B21, reverse those ordnance, A12, so that's going to be first row, second column is going to go in the second row, first column. So this one is going to go here, and then B22, I'll flip those, doesn't really matter, 22 goes here. What about the other entries of A? Notice we got these entries here, but what about these two? Well we can't ignore them. So we have an A31 and an A32 entry, which means that I have a B13 and a B23 entry. So B13, first row, third column is going to be the same as third row, first column. So third row, first column goes to first row, third column, that goes there, third row, second column, A32 goes to second row, third column, zero ends up there. So there's our transpose of the matrix. And this is still an algebraic operation, but visually why we call it the transpose is I've taken this matrix and I flipped it along its main diagonal.