Education Technology

# Activities

• ##### Subject Area

• Math: Algebra I: Matrices

9-12

45 Minutes

Derive™ 6

## The Action of a Linear Operator and an Introduction to Eigenvalues & Eigenvectors

#### Activity Overview

In this Derive™ activity, students use matrix multiplication as a tool for drawing the outlines of a simple stick house. They work in two dimensional space and show the results of multiplying a figure by 2 X 2 matrices.

#### Before the Activity

• See the attached DFW file for detailed instructions for this activity
• Print pages from the attached DFW file for your class
• #### During the Activity

Distribute the pages to the class.

• Set up a 2D plot window
• Initialize the state of Derive's random number generator by entering a random number
• Generate a 2 X 2 random number matrix
• Multiply the number with the matrix and save the resultant matrix as variable A
• Define a matrix H, of order, say, 10 X 2
• Run the ShowAction Derive function to obtain the transpose of matrix H
• Multiply the transpose of H on the left by A
• Examine the basic shape of the house (It is squashed and the bottom angles are not right angles)
• Define a matrix B to retain properties of the house, except its window orientation
• Run the ShowAction Derive function and note that the new house is the same house as the original one, but rotated
• Realize that the rotation is due to the trigonometric elements in the rotation matrix B
• Clear the plot window of plots, redraw the house using matrix A
• Run the ShowAction Derive function for successive powers
• Understand that the action of higher powers of A appears to be collapsing the house to a straight line
• Calculate the Eigen value λ for matrix A
• Check whether the determinant of A - λI = 0
• #### After the Activity

Students answer questions listed on the activity sheet.

Review student results:

• As a class, discuss questions that appeared to be more challenging
• Re-teach concepts as necessary