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appendixd_s

# appendixd_s - APPENDIX D SOLUTIONS TO PROBLEMS 0 1 6 2 1 7...

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APPENDIX D SOLUTIONS TO PROBLEMS D.1 (i) 016 21 7 2 0 61 2 180 4 5 0 5 36 24 300 ⎛⎞ −− ⎜⎟ == ⎝⎠ AB (ii) BA does not exist because B is 3 × 3 and A is 2 × 3. D.3 Using the basic rules for transpose, ( )() () ′′ = = XX X X , which is what we wanted to show. D.5 (i) The n × n matrix C is the inverse of AB if and only if C ( AB ) = I n and ( AB ) C = I n . We verify both of these equalities for C = B -1 A -1 . First, ( B -1 A -1 )( AB ) = B -1 ( A -1 A ) B = B -1 I n B = B -1 B = I n . Similarly, ( AB )( B -1 A -1 ) = A ( BB -1 ) A -1 = AI n A -1 = AA -1 = I n . (ii) ( ABC ) -1 = ( BC ) -1 A -1 = C -1 B -1 A -1 . D.7 We must show that, for any n × 1 vector x ,
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