ENEE241hw04

# ENEE241hw04 - ENEE 241 02 HOMEWORK ASSIGNMENT 4 Due Tue...

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ENEE 241 02 * HOMEWORK ASSIGNMENT 4 Due Tue 03/01 Problem 4A (i) (5 pts.) Review the concept of a rotation matrix (e.g., p. 60 in the textbook). Show that any matrix of the form r - s s r ( r, s R ) represents a counterclockwise rotation on the plane, preceded or followed by scaling (the scaling factor being nonnegative). Express the angle of rotation and the scaling factor in terms of r and s . (ii) (5 pts.) Consider the matrix A = cos(5 π / 24) - sin(5 π / 24) sin(5 π / 24) cos(5 π / 24) Does there exist a positive integer n such that A n = I (where I is the 2 × 2 identity)? If so, what is the smallest such integer? Explain . (iii) (5 pts.) Now let B = cos(3 π / 16) sin(3 π / 16) - sin(3 π / 16) cos(3 π / 16) Without explicitly computing matrix products, inverses, etc. , determine the matrix C such that A 2 CB 2 equals the identity matrix I . (iv) (5 pts.) Consider the matrices E = 1 - 1 0 1 1 0 0 0 2 and F = 1 / 3 - 2 / 3 - 2 / 3 - 2 / 3 1 / 3 - 2 / 3 - 2 / 3 - 2 / 3 1 / 3 What does E represent, geometrically? (The matrix F was introduced in the previous assignment.) Is it true that EF = FE ?

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