The matrix A = (13¹) is closely related to the imaginary number i. (a) Calculate A², A³, A4, expressing each as a multiple of either A or the identity matrix I. Infer the general pattern for A4, A4n+1, A4n+2 A4n+3, where n is an integer, and notice how this compares to powers of i (b) The exponential of a matrix is defined in terms of the power series eX = 1 + X + ² X ²X² + ²/1 ² Calculate e, evaluating the series to obtain a simple 2 x 2 matrix. Explain the result geometrically, noting the close analogy with e ·X³+.

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