(a) Prove that the product of two orthogonal matrices is orthogonal, and so is the inverse of an orthogonal matrix. What does this mean in terms of rotations?

(b) Show that (6) is an orthogonal transformation. Verify that it satisfies Theorem 3. Find the inverse transformation.

(c) Write a program for computing powers A^m (m = 1, 2, · · ·) of a 2 × 2 matrix A and their spectra. Apply it to the matrix in Prob. 1 (call it A). To what rotation does A correspond? Do the eigenvalues of A^m have a limit as m †’ ˆž?

(d) Compute the eigenvalues of (0.9A)^m, where A is the matrix in Prob. 1. Plot them as points. What is their limit? Along what kind of curve do these points approach the limit?

(e) Find A such that y = Ax is a counterclockwise rotation through 30° in the plane.

Data from Prob. 1

Are the following matrices symmetric, skew-symmetric, or orthogonal? Find the spectrum of each, thereby illustrating Theorems 1 and 5. Show your work in detail.

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0.6 0.8 -0.6 0.8