If C = a b c d and D = [ u v w z], then CD = DC yields 4 equations Ax = 0: CD+DC = 0 is

Chapter 4, Problem 4.15

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If C = a b c d and D = [ u v w z], then CD = DC yields 4 equations Ax = 0: CD+DC = 0 is 2a c b 0 b a+d 0 b c 0 a+d c 0 c b 2d u v w z = 0 0 0 0 . (a) Show that detA = 0 if a+d = 0. Solve for u, v, w, z, the entries of D. (b) Show that detA = 0 if ad = bc (so C is singular). In all other cases, CD = DC is only possible with D = zero matrix.

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