APPENDIX D SOLUTIONS TO PROBLEMSD.1(i) 0162172061218045053624300⎛⎞−−⎛⎜⎟==⎜⎝⎠⎝AB⎞⎟⎠(ii) BAdoes not exist because Bis 3 ×3 and Ais 2 ×3. D.3Using the basic rules for transpose,()()()′′′′′′==XXX X, which is what we wanted to show. D.5(i) The n×nmatrix Cis the inverse of ABif and only if C(AB) = Inand (AB)C= In. We verify both of these equalities for C= B-1A-1. First, (B-1A-1)(AB) = B-1(A-1A)B = B-1InB= B-1B= In. Similarly, (AB)(B-1A-1) = A(BB-1)A-1= AInA-1= AA-1= In. (ii) (ABC)-1= (BC)-1A-1= C-1B-1A-1. D.7We must show that, for any n×1 vector x,
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