 Remember that an elementary matrix consists of applying a single elementary row operation to the identity matrix. And as long as we don't switch rows, this will always give us a triangular matrix. And as long as we work top to bottom, these will always be lower triangular matrices. For example, suppose we want to find the elementary matrix corresponding to multiplying the second row of a matrix by negative 4 and adding it to the third row. So we'll begin with an identity matrix with at least three rows and apply the same row operation to get our elementary matrix. Strictly speaking, we should limit ourselves to a single row operation at a time. In practice, we won't distinguish between true elementary matrices and matrices that either multiply two or more rows by a constant or add multiples of one row to other rows. However, to retain at least a vestige of the idea that these are elementary-ish matrices, we won't combine both of these into a single matrix. We won't do both sets of row operations at the same time. So for example, let's find the matrix corresponding to multiplying the first row by 3 and adding it to a second row and multiplying the first row by negative 1 and adding it to a third row. So here the important thing is that our working row is the same in both cases. It's the first row and it's affecting rows below it, rows 2 and 3. And we'll proceed in the same way. We'll begin with an identity matrix with at least three rows and apply the same row operations, 3 times the first row plus the second and minus 1 times the first row plus the third. And this leads to what's called the LU decomposition. Suppose we row reduce a matrix A to U. If we don't switch any rows and work top to bottom, the elementary operations correspond to lower triangular matrices. So we can write LA equals U. Now remember the row echelon form of a matrix is an upper triangular matrix and L will be a product of elementary operations that correspond to lower triangular matrices and remember the product of lower triangular matrices is also a lower triangular matrix. So we know that L is a lower triangular matrix and U is an upper triangular matrix. And now if I multiply by L inverse I can write A as L inverse U. But remember the inverse of a lower triangular matrix is also a lower triangular matrix. And this means we can write A as a product of a lower triangular matrix L inverse as an upper triangular matrix U. For example let's find a LU decomposition for this matrix and then let's find the determinant and the inverse. So we use elementary row operations to row reduce to an upper triangular matrix. So our first set of row operations might be and we'll apply the same row operations to the identity matrix to get a quasi elementary matrix which we then apply to our matrix A. The next step in our row reduction will be and we'll get a corresponding quasi elementary matrix which is then applied to our result. Now we can multiply these two lower triangular matrices to get another lower triangular matrix and so here we have a lower triangular matrix applied to our matrix giving us an upper triangular matrix. So our last step is multiplying by the inverse of the lower triangular matrix which will be and if we multiply both sides by the inverse we get where our matrix is now written as a product of a lower triangular matrix and an upper triangular matrix. And remember the value of writing this as a product of triangular matrices is that some things are much easier to compute. Since the determinant of a product is the product of the determinant we can find the determinant of our matrix as the product of the determinant of the two triangular matrices and since they're triangular the determinant is the product of the entries along the main diagonal. And so our determinant is how about the inverse? Remember the inverse of a product is the product of the inverses in the reverse order and since these are triangular matrices we can find those inverses pretty easily we find and so the inverse will be