Wooldridge_5e_AppD_IM - APPENDIX D SOLUTIONS TO PROBLEMS 0...

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269 APPENDIX D SOLUTIONS TO PROBLEMS D.1 (i) 0 1 6 2 1 7 20 6 12 1 8 0 4 5 0 5 36 24 3 0 0 = = AB (ii) BA does not exist because B is 3 × 3 and A is 2 × 3. D.2 This result is easy to visualize. If A and B are n × n diagonal matrices, then AB is an n × n diagonal matrix with j th diagonal element a j b j . Similarly, BA is an n × n diagonal matrix with j th diagonal element b j a j , which, of course, is the same as a j b j . D.3 Using the basic rules for transpose, ( ) ( )( ) ′ ′ = X X X X X X , which is what we wanted to show. D.4 (i) This follows from tr( BC ) = tr( CB ), when B is n × m and C is m × n . Take B = A and C = A . (ii) 2 0 4 0 2 2 0 1 = 0 3 0 9 0 0 3 0 1 0 2 0 1 = A A ; therefore, tr( A A ) = 14. Similarly, 2 0 2 0 1 5 0 0 3 0 3 0 0 9 1 0 =
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