 Let's consider what happens when we row-reduce a square matrix A. Remember, we can perform elementary row operations by multiplying by an elementary matrix. And the elementary matrix corresponding to a row operation is the matrix produced when the row operation is applied to the identity matrix. For example, let's suppose E is the elementary matrix corresponding to multiplying the third row of a matrix by negative 2 and adding it to the fifth row find the determinant of E. Now, notice that we don't know the size of E except that it has at least five rows and columns. But since the third row is added to the fifth, only the fifth row changes. So we know the first four rows, and if we multiply the third row by negative 2 and add it to the fifth, we get the new fifth row. And if there are more rows, all other rows of the identity matrix are unchanged. Since this is a lower triangular matrix, the determinant is the product of the entries along the main diagonal. But since this was an identity matrix, those entries are all 1s, and so the determinant of this elementary matrix is going to be 1 as well. By a similar analysis, we find that the elementary matrix corresponding to switching two rows has determinant negative 1, the elementary matrix corresponding to multiplying one row by a constant c has determinant c. And we just determined that the elementary matrix corresponding to multiplying one row by a constant and adding it to another row has determinant 1. We can show that if E is an elementary matrix, then the determinant of Ea is the determinant of E times the determinant of A. But you should show it for row switching and for multiplying a row by a constant. Now this is also true if E corresponds to multiplying one row by c and adding it to another. This is a little harder to prove, and so we'll consider an example which we can generalize. And by we, I mean you. So we have our matrix and let E be the elementary matrix corresponding to multiplying the first row by 3 and adding it to the second row, let's find the determinant of Ea using only the defining properties of the determinant. So we'll go ahead and compute Ea, which will be. Now remember the determinant is linear in the entries of a row or column and so our determinant of Ea can be expressed as the sum of two determinants, but which ones? And the key here is only change the row that's changing. And since the second row was the one that changed, we'll keep the first and third rows the same. Since we'd like to relate the determinant back to the determinant of A, we'll make our first matrix A itself, which means the second row of the other matrix must be. So let's consider this second determinant. Now just that the second row of the second determinant is three times the first row and if you multiply one row of a matrix by a constant, you can factor out that constant. And so the determinant of the second matrix is three times the determinant with the three factored out. And since this matrix has two rows that are the same, each determinant is zero. So over on the left we have the determinant of Ea, over on the right we have the determinant of A plus zero, or just the determinant of A. Or since the determinant of E is one, we again have the determinant of Ea equal to the product of the determinants. So putting our results together, if E is an elementary matrix corresponding to a row operation on a square matrix A, then the determinant of Ea is the determinant of E times the determinant of A. Now Lather rinsed repeat to get the following. Let E1, E2, and so on be elementary matrices of the appropriate size where the product with A is equal to R. Then the product of the determinants is equal to the determinant of R. And this is an important result, and we'll see where it takes us next.