41_hwc1SolnsODDA

41_hwc1SolnsODDA - 5.21 Determine whether the following...

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5.21 Determine whether the following assertions are true or false. If true, explain why. If false, give a counterexample. (a) If A is any p × m matrix, and let B be a m × n , and the columns of B are all the same, then the columns of AB are all the same. (b) If A is any p × m matrix, and let B be a m × n , and the columns of B are all the same, then in each row of AB , all of the entries are the same. (c) If A is any p × m matrix, and let B be a m × n , and the rows of A are all the same, then the rows of AB are all the same. (d) If A is any p × m matrix, and let B be a m × n , and the rows of B are all the same, then in each column of AB , all of the entries are the same. SOLUTION (a) True. This follows from V.I.F. 3 : If for each j , the j th column of B is v , then for each j , the j th column of AB is A v . (b) True. In fact, (b) is just another way of stating (a) : The columns of a matrix are all the same if and only if the entries are all the same in each row. One can also show this directly, using
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