 So let's see if we can define the multiplication of two matrices. To do that, we'll go back to our transformation algebra. Suppose I have three transformations, A, B, and C. B corresponds to the transformation that takes the vector x to the vector y, and A corresponds to the transformation that takes the vector y to the vector z. The product A, B corresponds to applying B and then applying A so that this product A, B is going to take the vector x and send it to the vector z. And so we can define this product C equal to A, B, where C is going to take our vector x to the vector z. So let's make a few quick observations. In order for this to be possible, if B is a P by Q matrix, it will take vectors in F, Q to vectors in F, P, where F is our underlying scalar field. Since A takes a vector y to vector z, that means A must be an R by P matrix, so we can work on vectors in F, P, and send them to vectors in F, R. Which means that C has to be an R by Q matrix, which will take vectors in F, Q, and send them to vectors in F, R. And this gives us a very important result. If we have any hope of multiplying two matrices together, the number of columns in the first matrix must be equal to the number of rows in the second matrix. And the product is going to have the number of rows in the first and the number of columns of the second. And these considerations allow us to define matrix multiplication as follows. Suppose I have two matrices A and B, where the number of columns of A is equal to the number of rows of B. Then the matrix AB will be the matrix C, where the entry Cij is the sum of the component-wise products of the ith row of A with the jth column of B. For example, suppose we want to multiply the two matrices A and B. There's two products that we might be able to make A, B, and B, A. So remember that in order to be able to multiply two matrices, the number of columns of the first factor must equal the number of rows of the second factor. So since A has three columns and B has three rows, we can find the product AB. So the first row of A times the first column of B will give us the entry in the first row, first column, of the product. So we'll form the component-wise products, sum them, and that gives us our first entry, 5. We'll take the first row of A times the second column of B, find the component-wise products, and sum them, and that gives us our first row, second column entry. The second row of A times the first column of B will give us the entry in the second row, first column. The second row of A times the second column of B gives us the entry in the second row, second column. We'll take the third row of A times the first column of B to get the entry in the third row, first column. And finally, we'll take the third row of A times the second column of B to get the entry in the third row, second column. And this will be our matrix product AB. Since A is a 3 by 3 matrix and B is a 3 by 2 matrix, the product BA does not exist. Our definition of matrix multiplication also allows us to rewrite linear equations. For example, consider our system of equations and note that if we look at each equation, we can read the expression on the left as the product of the row of a coefficient matrix with a column of the variables. And so our system of equations can be rewritten as the matrix product coefficient matrix times column vector of our variables equals column vector of our constant terms.