 In this video, we present the solution to question number seven from practice exam number two for math 2270. We're given a two by two matrix A right here, negative one, one, one, negative one, which then factors into these four matrices right here. And we're supposed to describe what effect does the multiplication by A have on the plane R2 in terms of shear stretches, reflections, rotations, and projections. So we want to see the geometric effects of these matrices right here. Now, as the vector, if you multiply by A, the vector in play would actually be on the right. So it turns out that this matrix right here is the first one we're going to do second, third, and fourth right here. So that's the order here. We go right to left. So let's start with the first matrix. What type of matrix is that? So first, and we could list this, you know, something like this first, we have that this is a shearing matrix because the number is above the diagonal. This actually turns out to be a horizontal shear. So first we have a shear. We have a shear horizontally by a factor of negative one. That's the first one. The second one is going to be right here. Second. So what type of matrix is that? It looks like a diagonal matrix. We have zeros and ones along the diagonal. So this is not a, this is a singular matrix and they have a one in the first spot and zero everywhere else. This is a projection map. This is in fact going to be projection. This is projection onto, notice we're forgetting the y coordinate. We're going to keep the x coordinate. So this is projection onto the x axis. Okay. Then for our third matrix right here, this is a lower triangular matrix with a negative one below right there. So this is similar to step number one. Our third thing here, this is going to be a shear, but this is going to be vertically. When you talk about your shares, you do need to emphasize whether it's a horizontal or vertical shear. So we're shearing vertically by a factor of again, negative one. And then the last one, which would be right here. So this is the interchange matrix zero, one, zero, zero, one, zero. And so what this does is this is going to be reflection. This is going to be reflection across the diagonal line y equals x. There was no scaling, no stretches or compresses going on in here. No reflections across the x or y axis. And so these are the four, the four transformations that multiplication by a does make sure you go right to left. That's the most common mistake on this question number seven.