Help Linear Algebra

Question

Find formulas for the entries of M^n, where n is a positive integer. (Your formulas should not contain complex numbers.)

Answer 1

Writing M=-9I+7J. where I is the 2x2 identity matrix, and J is the matrix (J_{ij}) with J_{11}=J_{22}=0, J_{12}=-1, and J_{21}=1. Easy to see that J^2=-I, and obviously J and I commute. Thus the power of M becomes

M^n=(-9I+7J)^n which can be obtained using the binomial theorem. The formula for diagonal entries are the same, those for the off-diagonals differ by a sign. In a sense, this can be considered a version of complex numbers, with I being the unity 1, and J being the imaginary unit i (square root of -1). The details is left out as they are just elementary algebra.