# Geometry of 2x2 Matrix Multiplication with Intro Questions

The Geometry of 2x2 Matrix Multiplication In the applet below, dragging points X' and Y' will change the orientation of the transformed coordinate axis. It will also change the matrix that represents the transformation, as the columns of the matrix are the vectors X' and Y'. Even though point D and D' have different coordinates, they have the SAME coordinates with respect to their corresponding axis.

Notice that you can change the points in one shape. What happens to the other shape and why?

## Question 1

Where do the two shapes line up exactly? What is the matrix representation of this? Why do you think this works? Keep the shapes overlapping.

## Question 2

What happens when we change the location of X' and Y'? Try out several combinations and reflect below on any patterns you may see. Note: You may want to move along the axes first, then try other locations.

## Question 3a

Look at the matrix representation for the transformation. This tool uses matrix multiplication. Which elements in the matrices represent the coordinates for X' and Y'? How do you know?

## Question 3b

Which elements in the matrices represent the transformation? Explain which coordinate impacts each direction.

## Question 4

Look at the right-hand side of the representation for matrix multiplication. Write down some notes on possible rules for calculating matrix multiplication.

## Next Steps

Find a partner and discuss your answers to the questions above and any patterns you might see. Why does the graphical representation work?