What are Elementary Matrices Linear Algebra

In this video, we'll introduce elementary matrices. You can check the video chapters if you want to skip around. At this point, we've spent quite bit of time dealing with elementary row operations. We can multiply row by non-zero constant, interchange two rows, or add multiple of one row to another. Typically, we would not perform these operations on the identity matrix because the identity matrix is already perfect. We don't have any changes we want to make to the identity matrix. But in fact, performing these operations on an identity matrix gives way to the definition of an elementary matrix. We say that matrix is elementary if it can be obtained from the identity matrix by performing single elementary row operation. Let's see some examples. Here are three examples of elementary matrices. In this first example, we see that this is just the identity matrix, but after having the first row multiplied by three. So, single elementary row operation has taken place. We've just multiplied the first row by three, and thus, we have an elementary matrix. In this example, you can see that the first row and third row have been swapped. So, take the identity, swap the first and third rows, and you get this matrix here. Since it was obtained from the identity by performing single elementary row operation, this is another example of an elementary matrix. In this example, we can see that five copies of row four were added to row one. Because this was obtained from the 4x4 identity matrix by performing this single elementary row operation, this is called an elementary matrix. single row operation gets performed on the identity, that produces an elementary matrix. If matrix cannot be obtained from the identity by performing single elementary row operation, that matrix is non-elementary. Here are two examples of that. This matrix is not an elementary matrix because two row operations must have been performed on the identity to produce this matrix. We see that since there is an entry of two here, the second row must have been multiplied by two. But also, the first entry in the second row is one, which means row one must have been added to row two to produce this matrix. This is the result of two elementary row operations on the identity, so it's not elementary. This is another matrix which is not elementary. It cannot be obtained from the identity by performing single row operation. We see that two row operations were necessary to produce this matrix. First, row one must have been subtracted from row two to get this negative one. Then also, row one must have been multiplied by two to get that entry of two. So, this matrix is non-elementary. One of the most interesting things about elementary matrices is they give us way to view elementary row operations as matrix multiplication. If the elementary matrix results from performing certain row operation on the identity IM, and if is an by matrix, then the product EA is the matrix that results when the same row operation that produced is performed on In this way, an elementary matrix encodes the row operation that was performed to obtain it. And if we multiply another matrix by this elementary matrix, it will have the effect of that row operation being performed. Let's see an example. Here we have this 2 by 2 elementary matrix. We can tell that it came from the identity by adding two copies of row one to row two. That's how this entry became two. So, if we multiply this matrix by this matrix what should happen is two copies of row one get added to row two. The elementary matrix encodes the row operation that was performed to obtain it. So, the multiplication should produce 1 3 in row one because row one is not changed at all, but in row two, we should have two copies of row one added to it. So, it should be seven and then 6 + -4, so positive two. Let's actually do the matrix multiplication and see if this is what we get. Here is that matrix multiplication written out. You can see if we multiply by we capture the first row exactly by this 1 0 getting matched up with 1 5 and 1 0 getting matched up with 3 -4. That just duplicates the first row of But then, when we move on to the next part of the matrix multiplication, this 2 1 is going to have the effect of adding two copies of row one to row two. And so, we get that 7 2 as expected. If we had multiplied by this elementary matrix that results from swapping rows one and two, it would have the effect of swapping rows one and two of matrix So again, multiplication by an elementary matrix gives us another way to view the performing of elementary row operations. Once we understand that row operations can be performed by multiplying with elementary matrices, it's pretty easy to see that elementary matrices must be invertible because each one is obtained by performing single elementary row operation on the identity and every elementary row operation can easily be undone by another elementary row operation. And thus every elementary matrix can be undone by an inverse elementary matrix. For example, this 2 by 2 identity matrix could have its second row multiplied by 4, which would produce this elementary matrix. Then this could be undone easily by multiplying row two by 1/4. That would get us back to the identity. So the inverse of this elementary matrix would just be the elementary matrix that results from multiplying row two of the identity by 1/4. And that is this matrix here. So since each elementary matrix results by just performing an elementary row operation and every elementary row operation can easily be undone by another elementary row operation and each of these corresponds to elementary matrices, clearly elementary matrices have inverses. This guy times this guy is going to produce the identity. Another example, in this 2 by 2 identity matrix, we could swap rows one and two producing this elementary matrix. We could swap rows one and two again to get back to the identity. So this elementary matrix is actually its own inverse. Again, the elementary row operation corresponds to an elementary matrix, but so too does the elementary row operation which undos it. And so elementary matrices are invertible. One last example. This 2 by 2 identity matrix, we could take three copies of row two and add it to row one. But, we could easily undo that by subtracting three copies of row two from row one. Since this also has corresponding elementary matrix, this elementary matrix is certainly invertible. This matrix here is the elementary matrix which results from subtracting three copies of row two from row one in the identity matrix. And thus, this elementary matrix is the inverse of this one. And that's what elementary matrices are. Let me know in the comments if you have any questions. There is link in the description to video where we do some practice identifying elementary matrices. And if you're looking for more, be sure to check out my broader linear algebra course and linear algebra exercises playlists in the description.