 So let's introduce a couple of terms specific to matrices. We'd like to be able to talk about matrices in some sort of meaningful fashion. So let's begin with a couple of simple ideas. Given any matrix, I can describe its size by giving the number of rows and the number of columns, always specifying the number of rows first. So this matrix has two rows and three columns. So this will be a 2 by 3 matrix. On the other hand, this matrix has three rows and two columns, so this will be described as a 3 by 2 matrix. If I want to speak about the individual entries in a matrix, I'll give the location in terms of both the row and the column. So if my matrix is capital A, then A subscript ij, more properly i,j is going to refer to the entry in the i-throw jth column. For example, for the matrix shown, let's find its size and then find if it's possible A23, A32, A14, and A41. So the first thing we'll note is that this matrix has three rows and four columns, so this is a 3 by 4 matrix. Next, A23 refers to the entry in the 2nd row, 3rd column, and we find that that entry is 3. A32 is the entry in the 3rd row, 2nd column, which is 5, and A14 is the entry in the 1st row, 4th column, that's 0, and A41 is in the 4th row, wait a minute. There is no 4th row, so A41 does not exist. Typically we use the index notation to describe a matrix. So for example, let M be a 2 by 4 matrix whose entries are either 0 if i is not equal to j, or 1 if i is equal to j. So while this is a good description of M, we'll go ahead and fill it out in a more familiar form. And the first thing to note is that M is a 2 by 4 matrix, so it's going to have two rows and four columns. So we need to find M11, M12, and so on. So M11, because i is equal to j, is going to have value 1. M12, M13, and M14, because i and j are not equal, are all going to be 0. M21, because i is not equal to j, is also going to be 0. M22, since i equals j, is going to be 1. And M23 and M24, because i and j are not equal, are also going to be 0 again. We have a few more descriptive terms for matrices. A M by N matrix is square if M is equal to N. This just means we have the same number of rows as columns. We say that a matrix A is symmetric if and only if aij is equal to aji for all i and j. For example, this matrix has three rows and three columns, so it's a 3 by 3 matrix, and it's square. And we also see that aij is equal to aji for all i and j. So the entry in the second row, first column, is the same as the entry in the first row, second column, and so on. So that means the matrix is symmetric. And a simple statement you should be able to prove is that a symmetric matrix must be a square matrix. One way of looking at a symmetric matrix is that if we were to flip this matrix about its diagonal, we would get the same matrix. And this leads to the next idea. If we have M A N by M matrix, the transpose of M is a matrix A where aij is equal to mji. We designate this transpose as mt. And again, a nice, simple proof you should be able to do is to prove that a symmetric matrix is its own transpose. For example, suppose I have the matrix shown. Let's find its transpose. Now, while we define the entries of the transpose in terms of the entries of the original matrix, it's easier in math and in life to identify where we've come from before we think about where we're going. And in this case, we know where we started. So we know the entries of the matrix M, M11 through M23. To find the entries of the transpose, we'll just reverse those coordinates. So M11 is going to be the same as A11 where we've switched the row and column values. Now, because they're both one, it doesn't look like anything has happened. On the other hand, for M12 is going to be A21, M13 is going to become A31, and so on. So looking at our values, we see that M11, M12, and M13, those are the entries in the first row, will become A11, A21, A31, and those are going to be the entries in our first column. Likewise, M21, M22, and M23, the entries in our second row will become A12, A22, A32, the entries in our second column. And visually, it's as if we've flipped the matrix. So the rows in the original matrix become columns in the transposed matrix. There are also a couple of special matrices. So the N by M0 matrix is the N by M matrix whose entries are all equal to, no surprise, zero. And that by itself isn't particularly interesting, but it does allow us to define what's called a block matrix, also known as a partitioned matrix. And this is a matrix where we can partition it into blocks of zero matrices and non-zero matrices. So this matrix can be partitioned, can be cut into several pieces, where some of those pieces are zero matrices and the others are not.