Let's do another problem that involves square matrices. Here I have matrix a is 1, 0, 0 -1. b is -1, 0, 0, 1. Matrix x is a 2 by 1. It just has the two entries x and y. I want to compute 8 times x.
Remember you can do this multiplication because you have a 2 by 2 and a 2 by 1. Whenever the 2 inner numbers are the same, you can multiply the two matrices.
This matrix is going to be, x plus 0, x. It's going to be 0 minus y. Think about this means in terms of transformation. Imagine that this matrix represents the point x,y. What's x,-y? X,-y is the reflection across the x axis. This matrix creates the transformation of reflecting across the x axis.
Matrices can represent transformations. Let's see what matrix b does. Here I'm multiplying the matrix x by matrix b. We get -x plus 0, -x. 0 plus y, y. Again, if I think of the original x matrix, as the point x,y in the plane, what's represented by this matrix -x,y. It's the reflection in the y axis. That's the point that has coordinates -x y. This matrix reflects points across the Y axis.
Now what happens if we multiply a and b first and then multiply the result by x? Let's do that. We get -1 plus 0, -1. 0 plus 0, 0. 0 plus 0, 0. 0 minus 1, -1. I multiply -1, 0, 0, -1 by x,y. I get -x, 0 -x, 0 minus y. The point x,y has become the point -x,-y. If x,y is right here, -x -y is down here. Both coordinates have been reversed.
That has the effect of reflecting across the y axis and then reflecting across the x axis. So it's like we got both transformations when we multiplied the two matrices. Multiplying matrices can give you a sequence of transformations.