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Question: 2 sometimes matrices appear as the argument of a transcendental...

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2. Sometimes matrices appear as the argument of a transcendental function. Let M be defined as M- -1 Let x E R. We can calculate A = elf as follows: (a) Compute M2, M3 and M (b) From the results of (a), infer what M must be for any positive integer n. (c) Write out the Taylor series expansion for e*; ie ez = 1+2+-+-+-+-.. 3!4! (d) Substitute z xM into the above Taylor series and carry out the appropriate operations. Replace the first term, I, by the 2 × 2 identity matrix . Note that A exM İs a matrix! (e) Write out the first few terms for each element of A () Do you recognize the resulting series for each element? If so, replace the infinite sum in each element by its function. (g) What are the entries in the matrix A when x = π/2? Useful Information: sinx= x-3, -+-+ S! cos x = 1--+-+-.. 2! 4!

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