 Summarizing what we've learned so far about multiplication by two by two non-singular matrices, we've seen there were a couple different transformations we could do. We could get a shear when we multiplied by a unit upper triangular or lower triangular matrix. Be aware that shearing maps, shearing maps actually coincided with the replacement elementary matrices that we'd seen before, okay? Then we saw that if you were to have a reflection, we could get reflections, which reflections, well, if you reflected across the line y equals x, that corresponded to the only interchange elementary matrix applicable for two by two matrices. But then also, if you had reflections to correspond to multiply by negative diagonal matrices, which that would be an example of an elementary matrix of scaling type. And then the last transformation we've discussed so far would be then dilation scaling of some kind, which as stretching and compressing corresponds naturally to elementary matrices of scaling type. And so you see all of them right here, replacements, interchange and scaling if we then put positive and negatives together. This shows us that the three elementary matrices correspond, multiplied by them correspond to these three geometric types of transformations. And so this leads to the following observation right here. If A is a two by two non-singular matrix, so it has an inverse, then the matrix transformation associated to multiplying by A can be decomposed into some composition of shears, reflections and stretches and compressions like we saw before. And the basic idea here is that the matrix A can be factored into a product of elementary matrices since it's non-singular. And each of these elementary matrices can be represented as reflection shearing or stretches and compressions. And so I wanna show you an example of this right here. Consider the matrix 0, 1, 2, 1. So we can show that this matrix is in fact non-singular. What we would do is, well, we would first probably interchange the matrix, the rows right here. So here's an interchange matrix right there. Then when you interchange the rows, you'll end up with 2, 1, 0, 1. The next thing to do is, well, since the matrix is already an echelon form, we would probably, we could divide, there's a couple of things we could do. We could divide the first row by two. So that would then, dividing the row by two would turn the matrix into this one right here, 1, 1, 1, 1, 0, 1. Which if since we divided by two, we would take scale the first row by two, it's inverse. And so you can see that by putting this matrix into echelon form, we found a factorization actually into elementary matrices like so. So this was our interchange matrix, this was our scaling matrix, and this is a replacement, a backwards replacement matrix, upper triangular, upper unit triangular matrix. So this is going to be a factorization into elementary matrices of A. Now, if we think of this as a geometric transformation, what does this matrix, what does multiplying by A do to a generic vector? So let's think about that for a second. Let's take a generic vector x times, or x and y right here. If we were to multiply A by x, y using this factorization, this matrix on the far right is actually gonna be the first matrix that acts on x, y. So this would be our first matrix. And so looking at that matrix, this is a shearing matrix, it's upper triangular. So that means it's gonna be a horizontal shear by a factor of one half. So this is our first matrix. It's gonna have a horizontal shear by a factor of one half. The next matrix to interact with the point would then be the matrix in the middle, this matrix two, zero, zero, one. This is a scaling elementary matrix. It's gonna scale the first row by a factor of two. So this is gonna have the geometric effect of horizontally stretching the graph by a factor of two. And then the third matrix to interact with the point will be the one on the far left, this interchange matrix, zero, one, zero, zero, one, one, zero there. And so we've seen that before that this interchange matrix is the effect that you reflect across the line, y equals x. So it's important to realize that when you work with these geometric matrices here, if you wanna figure out what does this thing do geometrically, you can decompose the matrix A into a product of elementary matrices. Cause we know how each of these elementary matrices affect the plane. And then reading it right to left, reading it right to left, because again, the matrix on the far right will interact with the vector first. Reading it right to left, you then see what the transformations are gonna do. So given any matrix, let's summarize, factor it into elementary matrices and reading it right to left, we can see how that matrix will affect the plane geometrically.