 So let's think about this. Given any coefficient matrix A, we can reduce it to row echelon form by switching to rows, multiplying a row by a constant, or adding a multiple of another row to a given row. These can be performed by left multiplication by an elementary matrix, so we can represent the process as a product, where EI is an elementary matrix and R is the row echelon form of A. And since the determinant of a product is a product of the determinants, this suggests an easy way to find the determinant. For example, let's find the determinant of this 4 by 4 matrix. So if we wanted to row reduce this matrix, the first step in our row reduction might be multiplying the second, third, and fourth rows by 3. And to find the corresponding elementary matrix, we do the same thing to the identity matrix. Strictly speaking, this should be done by 3 elementary matrices, but we'll combine all the row operations into a single matrix. So if I were to take the identity matrix and apply the same row operations I'd get, so the first step in our row reduction could be accomplished by left multiplication by this matrix, and this is not just triangular, it's actually a diagonal matrix, and the determinant is going to be the product of the terms along the diagonal, which will be, now going back to our matrix, our next step would be to add a multiple of the first row to the second, third, and fourth row. And so those elementary operations would be, and again if we do the same thing to the identity matrix we get, and so the second step in our row reduction can be performed by left multiplication by this matrix, and notice this is a lower triangular matrix and its determinant will be, again our matrix looks like this, and so now we can multiply the third and fourth rows by 14, and this step in the row reduction can be accomplished by multiplying by this matrix with determinant 196, the next step in our row reduction with corresponding matrix and determinant 1, and similarly for the last two steps in our reduction to row echelon form, and our corresponding matrices will be, so remember the determinant of a product is the product of the determinants, so on the left hand side the determinant of that product is going to be the product of the individual determinants of our matrices performing the row operation times the determinant of A. On the right hand side we have the determinant of R, which is the row echelon form of our matrix. You might think that this hasn't really helped us because we still have to find the determinant of a 4 by 4 matrix, but the reason this is useful is that our final matrix is an upper triangular matrix, and since the row echelon form is upper triangular its determinant is easy to find, and now we can solve for the determinant of A.