 Let's see if we can introduce other operations on matrices. So suppose I have a matrix P and a scalar A. What can we make out of A times the matrix P? Since P is a matrix, well, it's actually a linear transformation. So maybe we should think about P as a linear transformation instead. We should consider what A P as a linear transformation does to some vector V. And so, since P is a matrix, I mean linear transformation, then we know something about what P does to A V. Because it's a linear transformation, we know that P applied to A V is A times P applied to V. And what's worth noting is that this A times P applied to V looks an awful lot like this A P applied to V. And so it would be very, very, very, very, very, very, very, very, very, very convenient if A P applied to V were the same as A times P applied to V. And if that were the case, then suppose P applied to V gives us the vector U, then A P applied to V should give us the vector A U. And that gives us a starting point in defining the scalar multiplication of A times the matrix P. And so we'll make the assumption that A P applied to V is the same as A times P applied to V. So let's consider this. As a matrix, I mean as a linear transformation, P is going to take some vector V to some vector U where the entries of P correspond to the coefficients in the formulas that give us the components of the vector U. If we want the matrix A P to take our vector V and send it to the vector A times U, then we want those coefficients to give us the components of A U. So if I want to get the components A U 1, A U 2, and so on, what do I have to do to my formulas? And the simplest thing to do is to multiply each of our formulas by A. And once we have the formulas, we can recover the transition matrix. So this suggests that the matrix A P is going to be the matrix P with every entry of that matrix multiplied by A. And so this gives us our definition for what the scalar multiple A times the matrix P is going to do. The matrix A P is going to see the matrix P with every entry multiplied by the scalar A. So for example, if I have the matrix 3, 1, 5, 1, 0, 4, I can find 3M by multiplying every entry of the matrix by 3. So my first row, I'm going to take every entry and multiply it by 3. And likewise for my second row, I'll take every entry and multiply it by 3.