Show that any matrix of the form (r, 8 R) represents a counterclockwise rotation on the...

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Show that any matrix of the form (r, 8 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= Does there exist a positive integer n such that A" = I (where I is the 2 x 2 identity)? If so, what is the smallest such integer? Explain. (iii) (5 pts.) Now let 1 --0 E- cos(3x/16) sin(3/16) Without explicitly computing matriz products, inverses, etc., determine the matrix C such that A CB equals the identity matrix I. (iv) (5 pts.) Consider the matrices cos(5m/24) -sin(5/24) sin(5/24) cos(57/24) B= -1 0 1 0 0 2 1/3-2/3-2/3 and F= -2/3 1/3-2/3 1/3 What does E represent, geometrically? (The matrix F was encountered in Problem 7). Is it true that EF = FE? Show that any matrix of the form (r, 8 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= Does there exist a positive integer n such that A" = I (where I is the 2 x 2 identity)? If so, what is the smallest such integer? Explain. (iii) (5 pts.) Now let 1 --0 E- cos(3x/16) sin(3/16) Without explicitly computing matriz products, inverses, etc., determine the matrix C such that A CB equals the identity matrix I. (iv) (5 pts.) Consider the matrices cos(5m/24) -sin(5/24) sin(5/24) cos(57/24) B= -1 0 1 0 0 2 1/3-2/3-2/3 and F= -2/3 1/3-2/3 1/3 What does E represent, geometrically? (The matrix F was encountered in Problem 7). Is it true that EF = FE?

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i To show that any matrix of the form A E 0 R represents a counterclockwise rotation on the plane preceded or followed by scaling we can analyze the m... View the full answer