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Square Matrices - Problem 2

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Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

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.

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