39_hwc1SolnsODDA

39_hwc1SolnsODDA - B can be free modied without aecting AB...

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5.9 Let A be an m × n matrix, and B be an n × p matrix. If B has at least one zero column, can it ever be the case that AB has no zero columns? Explain why not, or give an example. SOLUTION No. By V.I.F. 3 , which is the definition of matrix–matrix multiplication, (column j of AB ) = B (column j of B ) . Therefore, if (column j of B ) = 0, (column j of AB ) = B 0 = 0 . 5.11 Let A be the matrix A = 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 . Let B be any other 4 × 4 matrix. (a) Which rows of B , if any, can be freely modified without affecting the product AB ? (b) Which columns of B , if any, can be freely modified without affecting the product AB ? SOLUTION (a) For any vector v = a b c d , A v = a b 0 0 . That is, the result is independent of the third and fourth entries of v . Now, by V.I.F. 3 , if v is the j th colum of B , then A v is the j th column of AB . Therefore, the entries of AB come out the same no matter what the third and fourth entries in each column of B are. In other words, the third and fourth rows of
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Unformatted text preview: B can be free modied without aecting AB . (b) By the rst part of Theorem 13, or what is the same, V.I.F. 4 , ( AB ) 1 ,j = (row 1 of A ) (column j of B ) . Since (row 1 of A ) 6 = 0, by changing (column j of B ), we can change the value of A 1 ,j . (The same sort of argument wold apply to A 2 ,j as well). Therefore, none of the columns of B can be freely modied. 5.13 Let C be a 2 by 2 matrix such that C h 1 2 i = h 2 1 i and C h 2 1 i = h-1 1 i Using the given information, nd 2 2 matrices A and B so that CA = B , and then solve for C . SOLUTION By V.I.F. 3 , if A = [ v 1 , v 2 ], then CA = [ C v 1 , C v 2 ]. Therefore, we can use the given information if we take v 1 = 1 2 and v 2 = 2 1 , 21 /september/ 2005; 22:09 40...
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