[MUSIC] For those of you who don't know about matrices or about matrix algebra or for those of you who might have forgotten their linear algebra from university, I want to give you a short lecture on the information you need to know about matrices to understand a couple of lectures in this course, mainly those leading up to Cassini's identity. So we start with the very basics. Capital A here is what's called a matrix. And it is what we call a two by two matrix. That means it has two rows, a b, c d, and two columns, ac and bd, so we say it's a two by two matrix. X here, written in bold face, is what we call a vector, or we also call it a column matrix. For the two component case, we write the column matrix as x y. So it has one column and two rows, okay. In this course, these are the sizes of the matrix that we need to deal with. No larger matrices than this. So if we can understand everything we need to know about two by two matrices, then we can understand all of the material in the course that deals with matrices. So in this video, I want to show you two simple operations, addition and multiplication. So let's see how addition works. So if we take two two by two matrices, a b c d, e f g h, and add them together, the result is very simple. It means that you add the element by element. So in this first row, first column element, you add it to the first row, first column element a plus e and it goes in the first row, first column element. So a plus e, b plus f, c plus g, d plus h. So matrix addition is just element by element. The only requirement is that you're adding matrices of the same dimensions. So here, we're adding two, two by two matrices. What about matrix multiplication? Matrix multiplication is a little more complicated, but not that difficult. For the two by two case, we can multiply a two by two matrix times a two by one column vector, okay, a two by one matrix, which is a column vector. And the multiplication proceeds by going across the rows of the first matrix and down the column of the second matrix. So the first element here will be a times x plus b times y. And the second element here, in the second row, will be c times x plus d times y. So the first row times the first column gives us the first row, first column. And the second row times the first column gives us the second row, first column, okay. So that's a matrix times a vector. What if we multiply two two by two matrices? So we're multiplying a b c d times e f g h. Well, we always go across the rows of the first matrix and down the columns of the second matrix. So we have a times e times b times g, and that gives us the first element here. So the first row multiplied against the first column gives us the element in the first row and first column. Then we go ab multiplied against fh, so af times bh. That goes in this element. So the first row times the second column gives us the element in the first row and second column. Then we do cd times eg. That will give us ce plus dg, and that goes here. And then we go cd times fh. So cf plus dh. And that goes in this second row, second column element. So in multiplying matrices, you go across the first row. You go across the rows in the first matrix and down the columns in the second matrix. And that's how you multiply. Since we're only doing two by two matrices in this course, this is all you need to know about addition and multiplication. In the next video, I will try to introduce the concept of determinants.