 An important type of special matrix is called a triangular matrix. Since every matrix we've seen has been rectangular, you might wonder how we can talk about a triangular matrix. So suppose I have a matrix m. If the entries of m are zero whenever i is less than j, then m is said to be a lower triangular matrix. So for example, in a 4x4 matrix, anytime i is less than j, the entry is going to be zero. And that means all of these entries are going to be zero, and our matrix looks like this, where the only non-zero entries are in this lower triangle. Similarly, we can define an upper triangular matrix where the entries are zero if i is greater than j. And so in this matrix, all of these entries are going to be zero, and the non-zero entries are in this upper triangle of numbers. And finally, as a special case, if our entries of the matrix are zero anytime i is not equal to j, then m is a diagonal matrix, and that's because any of the entries that are not along the main diagonal are going to be zero. So let's try some proofs. How about this one? The sum of two upper triangular matrices is also an upper triangular matrix. Well, maybe it's true, maybe it isn't, let's find out. So again, part of the reason that proof is valuable as a study tool for higher mathematics is that it forces us to review our definitions. So we want to talk about triangular matrices and the sum of two matrices. So let's see what happens. If I have two upper triangular matrices where their sum is equal to something, from the definition of the sum of two matrices, we see that this sum matrix c will have entries aij plus bij. So what can we do with this information? Since we know that the two matrices are upper triangular, our definition says that if i is greater than j, then aij and bij are both going to be zero, so cij will also be zero. And that means c itself will also be an upper triangular matrix. How about the product of two upper triangular matrices? It's vitally important to remember that an example is not a proof. However, it's helpful to consider examples to guide our proof. So let's take two upper triangular matrices and multiply them together. And if we do that, we see that we do actually get an upper triangular matrix. So this suggests that our statement might actually be true and be worth trying to prove. But again, an example is not a proof. Now, since we're trying to prove something about the product of two upper triangular matrices, we might need to pull in our definition of product of two matrices and upper triangular matrix. Because we multiply two matrices by taking the rows of the first matrix and multiplying by the columns of the second matrix, we might consider what happens when we take, for example, the second row of this matrix and multiply it by each column of the other matrix. And the thing that you notice is that we have this lineup of the zeros in the rows with the zeros in the columns. And because of that, in some cases, all of our entries are going to be zero. So let's see if that happens generally. So suppose a and b are two upper triangular matrices with c, the product of a, times b. So the entry in the i-th row j-th column of our product matrix is going to be the sum of the component-wise products of the i-th row of a with the j-th column of b. And so the entries in the i-th row of a are going to be a i1, a i2, and so on. And the entries in the j-th column of b are going to be b1j, b2j, and so on. We'll multiply them and add them together to get our entry in the i-j-th position of the matrix c. Now if we look at our example, we see that each row of a begins with several zero. And this is because a is upper triangular, aik will be zero if i is greater than k. And so the first couple of terms of this sum will drop out. And in fact the first place where we might have a non-zero term is going to be the i-th entry of the i-th row. Since our definition of upper triangular matrix has to do with the entries when i is greater than j, we might consider what happens if i is greater than j. If i is greater than j, then bi-j is equal to zero. And so is bi plus 1j, and so on. And so all of these terms in the sum are also going to be zero. So this sum will also be zero, and our matrix entry will also be zero. And now c fits the definition for being upper triangular because cij will be zero whenever i is greater than j.