 So it's important to talk about how given some rings we have how we can construct new rings from old rings And in this video I want to talk about the ring of matrices So in linear algebra and also in in group theory in this lecture series We've talked a lot about matrices groups of matrices. We can talk about groups of matrices under addition We can talk about groups of non singular matrices under multiplication but if we combine matrix addition and multiplication together we can actually form rings of Matrices so for example, if you take the set of n by n real matrices or the set of n by n Complex matrices these will make rings under the usual matrix addition and matrix multiplication These are also rings with unity since the identity matrix I Sub in so this would be that the matrix with ones along the diagonals and then zeros everywhere else, right? The identity matrix is given that name because it's the multiplicative identity And therefore these rings of matrices the real matrices and complex matrices These are rings with unity on the other hand though These rings are not commutative since matrix multiplication is Non-commutative if I just choose two arbitrary matrices We cannot expect that a times b is equal to b times a now. There are some special cases that a times b is equal to b times a with matrices here when we say something like non commutative What we're saying is as we're not saying it's never commutative It just means there's no guarantee of commutivity of commutivity here. There are and then for matrices There are some matrices for which a times b does not equal b times a it's easy to find this If you just pick two random matrices, you most likely will find out that they do not commute with each other So these are non commutative rings with unity, but matrix rings are very important rings to study in ring theory in fact If you have any ring whatsoever because there's nothing particularly special about the ring of real numbers or complex numbers here If r is any ring, then we can form the matrix ring which will denote m sub n by n of r to be the set of all n by n matrices with entries coming from the number r from the ring r Now addition of matrices is defined analogously to where terms are added together component wise Okay, so for example, if we have any ring, you know, we'll do two by two matrices for examples. You have any ring a b c d we'll do something like that. I'll call it like a one b one c one d one If you add this to the to the matrix a two b two c two d two Where here we're assuming that a i b i c i and d i these are all just elements of a ring When it comes to adding that matrices together Well, what you're going to do is you're going to add together the one one position So you get a one plus a two you add together the one two position So you get b one plus b two you're going to add together the two one position Which gets c one plus c two and you're going to add together the two two position Which is d one plus d two Like so in which case to add together matrices just by Component wise you just have to add together the components You have to add together the a's the b's the c's and the d's And so if you want to construct matrix addition all that requires of your scalar is that you can add scalars If you can add scalars you can add matrices great rings have addition so we can add together the we can add together the The matrices now what if we want to think about like multiplication? What does it mean to multiply together two matrices? Well by the usual rules of multiple matrix multiplication We take like the first row times the first column this finger multiplication here You're going to get a one a two plus b one c two Then you would take the first row times the second column You're going to end up with a one b two Plus b one d two Then you're going to take the second row first column You end up with c one a two plus d one c two And then lastly you can take the second row times the second column You're going to end up with c one b two plus d one d two If you follow the usual rules of matrix multiplication, you would end up with this product Now this right here works for any ring Because in order to do this product if you take a row times a column We have to be able to multiply together elements right we have an a one times an a two We have to know what that means we have to have a b one times a c two We have to multiply and then we have to add which in a ring we can multiply and add so in a ring If our scalars are from that ring then we can multiply together matrices now I have to caution you here that The ring might be non commutative. Okay, the the the the the elements themselves might not commute We know that matrices in general don't commute But what if the scalars themselves don't commute? Well, that's okay because when it comes to this when these when you come to these products Notice how it's always one two one two one two one two You always when you have your products you always have the scalar from the first matrix times the scalar from the second matrix Plus a product where you have a first scalar second scalar plus a product from a first scalar second scalar This example I did two by two matrices But you can see how very naturally this could be brought into three by three four by four any end by end matrix So even in a non commutative setting we can make matrix multiplication be well defined Just be very cautious of the order of operations here If the ring r is a ring with unity then Then um, this matrix ring will have unity as well because it'll just be the identity matrix In the usual sense. So if r is a ring with unity, then this will be a ring Then the matrix ring will be a ring with unity now even if the ring though r is commutative If you have if n is greater than one that is you have two by two three by three four by four Your matrix ring is not going to be commutative even if the ring is but important to emphasize here that the matrix ring Is well defined even if the ring is not commutative Only thing we require for the ring is that the addition makes an abelian group And that we also need multiplication to be associative and distributive. We don't need commutivity We don't need identities if you have a multiplicative unity then so will the matrix ring But commutivity will never be guaranteed when you have matrix rings