 Now that we have some matrix arithmetic at hand, let's take a closer look at elementary row operations. Let A be an M by N matrix. Find, if possible, a matrix R that corresponds to the elementary row operation of multiplying the first row of A by some constant C. To solve this problem, we need to set up a system of equations. First, we know A is an M by N matrix and R A must also be an M by N matrix. So if we go over our rules for matrix products, this means that R must be an M by M matrix. Now by matrix multiplication, the entry in the first row, first column, must be the sum of the component-wise products of the first row of R with the first column of A. And we'd like that to be C times A11. And this has to be true regardless of the values of the entries of the first column of A. And that means the only way that this can be guaranteed is to make R11 to be equal to C and R1I to be 0 for all the other entries. Now if this is the first row of our matrix R, then when we multiply this by the second column, we'll get the first row, second column entry, and that's going to be C times A12, which is exactly what we want. And if we do this for all the other entries of the first row, we see that making this, the first row of R, will in fact multiply the first row of A by the constant C. What about the entries in the second row? Well, these will be found by multiplying the second row of R by each of the columns of A. So the second row, first column entry, is going to be... And this has to be equal to A21 regardless of the values of the entries of A. And the only way we can guarantee that that will happen is R22 has to be 1, and every other entry in that second row of R must be equal to 0. And if we take that as our second row of R, then the entries in the second row of R will be unchanged. This generalizes, and that allows us to find the remaining rows of the matrix R. Trust me, I'm on the internet. Or verify for yourself. So again, an important thing you want to do as a mathematician is to, more or less as a matter of habit, ask yourself the question, how can we generalize this? Well, first of all, we have to produce a name for it. And so this matrix, which can be used to perform an elementary row operation, is referred to as an elementary matrix. And the thing to notice here is that this elementary matrix is the identity matrix with C instead of 1 in the first row. And a good habit to get into as a mathematician is to ask how this generalizes. In this case, what changes if we want to multiply the kth row of A by our constant C? And if we examine what actually changes, we come to the following theorem. The elementary matrix correspond to multiplying the kth row of a matrix by C is the matrix R, whose entries are going to be C if i is j and j is k, 1 if i is j and j is not equal to k, and 0 everywhere else. And again, this is just going to be the identity matrix with a C in the kth row instead of a 1. Another way to generalize this problem is by noticing that we tried to find row operations by multiplying on the left. What happens if we try to reproduce a row operation by multiplying on the right by some matrix? So again, let's let A be a matrix. And if possible, I want to find a matrix R where if I multiply on the right, I will multiply the first row by some constant C. So again, I'll set up our system of equations. And again, we'll want to make sure that our first row first column entry is correct. So we get this entry by multiplying the first row of A by the first column of R. And this tells us that R11 will have to be C and Rn1 will have to be 0. Since we now know the first column of the second matrix, we can find the first column of the product. So let's multiply the second row of A by the first column of R and get what will be the entry in the second row first column. Unfortunately, when we do that, we find that the second row first column entry is going to be C times A11, which is not what we want it to be. And what that says is that we can't find a matrix R that multiplies on the right and gives us the matrix form by multiplying the first row of A by a constant. And so no such matrix exists. Another elementary operation is multiplying a row by a constant and then adding it to another row and then replacing that other row. So let's see if we can find an elementary matrix that corresponds to that row operation. So again, we'll set up our matrix product and the first entry of the second row will be and now if we compare coefficients, we see that R21 has to be equal to C and R22 has to be equal to 1 and all the other entries in that second row of R will have to be equal to 0. And so this choice gives us all entries in the second row and since every other entry in the product of the two matrices is unchanged, all the other entries are going to correspond to the identity matrix. Trust me. And again, we can generalize this result if we want. The elementary matrix corresponding to multiplying the kth row of a matrix by C and then adding it to the mth row of the matrix is the matrix R where the entries of R are going to be 1 if i is equal to j. So it looks just like the identity matrix with the exception that we're going to have an entry C if i is equal to m and j is equal to k. And one thing that's worth noting about this is that as long as we don't do row interchanges and as long as we work from top to bottom, these elementary row operations are going to correspond to lower triangular matrices.