 So let's take a look at what we might think about as the matrix zoo. There's a number of special terms that we use and special types of matrices that are important. So matrices are fairly important and we do want to be able to distinguish between certain types of matrices. So let's go ahead and take a look at some of those forms. The zero matrix is the matrix with surprise, a matrix where all of the entries are equal to zero. Square matrix, again no surprise here, has the same number of rows and columns. And so, for example, we might take the matrix A, this is a zero matrix, this is a zero matrix, this is a zero not a zero matrix. And we might say that, well, A is both a square matrix because the number of rows and the number of columns are the same. It's square and zero, B is a zero matrix but it's not square because it has different numbers of rows and columns and then C is a square matrix. Same number of rows and columns but it's not a zero matrix. A particularly useful classification is diagonal and triangular matrices. So if I have a diagonal matrix, all of the entries are going to be zero except possibly those along our main diagonal. If I have an upper triangular matrix, that's going to be all entries below the main diagonal are zero. So all of our non-zero entries are going to be concentrated above the main diagonal. And in the lower triangular matrix, no surprise, all of the entries below the main diagonal are the non-zero entries. So again, here I have a matrix A, here's my main diagonal, that matrix. And so here, my main diagonal, everything's above it. So A is upper triangular, for B, here's my main diagonal and everything's below that. So this is lower triangular and for C, just the diagonal has non-zero entries. So this is a diagonal matrix. So a useful form to have a matrix in is known as row echelon form or reduced row echelon form. And that emerges as follows. The matrix is in row echelon form if in each row the first non-zero entry is a one and everything below that entry is going to be a zero. Now, if we also have all the entries above the leading one, if those are also zero, then the matrix is in reduced row echelon form. So for example, a second row of matrices, so matrix A, first non-zero entry is a one and zeros are below it. Second row, first non-zero entry is a one and there's zeros below it. Third row, first non-zero entry isn't anything, there is no non-zero entry. So trivially we've satisfied the first non-zero entry is a one. So this matrix A is in echelon form, row echelon form. For B, our first non-zero entry of the first row is a one below it, though we have a non-zero entry. So B is not in any sort of row echelon form. And then for C, first non-zero entry is a one, zero is below, first non-zero entry is a one, zero is below, and above. First non-zero entry is a one, zero is below, trivially nothing's below it, but also above it. So C is actually in reduced row echelon form. Now probably the most important matrix we're going to run into is the coefficient matrix. And so the idea here is maybe I have some sort of system of N linear equations in M variables. So by variables, typically we call them x, y, z, and so on, but because we may have a lot of variables, we're going to index it x1, x2, and so on. So here's our variables, M variables, and the coefficient matrix is going to be the matrix defined by AIG, the component, the ij component of the matrix A is going to be the coefficient of the jth variable, that's our column number, in the ith equation, that's our row number. And we do this for a very specific reason that will become apparent when we do the few examples. So one important proviso here is that our equations have to be in standard form, where we have all of the variables on one side of the equals symbol and the constant on the other. So for example, I have my system of equations, I have one, two equations in x, y, w, z. I have four variables, x, y, z, and w, and so I have the system of two equations in four unknowns. And so first of all, before I do anything, I have to put the equations in standard form, so I have to rewrite that second equation. So I've got to get all of my variables over onto one side. And now I have my two equations in standard form. Now, while the order of the equations is pretty clear, first equation, second equation, and it also doesn't actually make a difference whether I call this the first equation and this one the second, we do have to specify the order of the variables. So we have to choose an order for our variables and so maybe my order will be x, y, z, and w in our standard mathematicians version of alphabetical order. And if we do that, our coefficient matrix is going to be, well, let's take a look at that. Well, again, the aij entry of the coefficient matrix is going to be the coefficient of the jth variable in the i-th equation. i is the row number, j is the column number. And so the reason we do it this way is this allows us to go row by row and write down the coefficients of the variables. So my first row is going to correspond to my first equation, and the first, second, third, and fourth columns are going to correspond to the coefficients of my first, second, third, and fourth variables. So a11, that's the coefficient of the first variable in the first equation, so that's going to be 3. a12 is going to be the coefficient of the second variable in the first equation, that's negative 4. a13 is going to be the coefficient of the third variable in the first equation, and that third variable doesn't exist, so that's coefficient is 0. a14 is the coefficient of the fourth variable, w, in the first equation, that coefficient is 8. And likewise I can find a21 through a24. Those are the coefficients of the variables, 1, 2, 3, and 4, the coefficients of the variables in the second equation. So that'll be 1, negative 3, negative 6, negative 2. And there's my coefficient matrix. Now the thing you might notice is that when we form the coefficient matrix, we lost the constants of the equation, and we might actually care about those constants because, well, among other things they allow us to solve the equation, and we can restore them by producing what's called the augmented matrix. Now what this really means is that I'm taking a matrix and then I'm making it a slightly larger matrix by augmenting it by adding some, by including some additional, in this case, column or columns. So, for example, my system of equations 3x minus 4y plus aw equals 5, this original system of equations, I produce the coefficient matrix by picking off the coefficients of the variables, but I dropped out the constants 5 and 1, and so I can restore them by including an extra column. So there, these constants here get shown up as constants over here, and again they are coordinated with the corresponding rows. So remember our first row corresponded to our first equation, so this value 5 has to be that constant 5, and likewise second row corresponds to the second equation. And this produces what we call an augmented matrix.