Linear Algebra with Applications (3rd Edition)

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ma253 , Fall 2007 — Problem Set 4 This problem set deals with linear transformations and matrices, as be- fore, with special attention to the issue of invertibility and to matrix mul- tiplication and its meaning. Plus, just a touch on kernels and images. As usual, solve all the problems, then write up and turn in those marked with an asterisk. This assignment is due on Wednesday, October 10 . 1. Problems from the textbook: a. Section 2 . 3 , problems 1 , * 2 , * 4 , * 12 , 29 , * 30 , 31 , * 34 , * 43 . b. Section 2 . 4 , problems 31 , * 32 , 33 , * 36 , * 38 , * 44 , 45 , * 46 , * 86 . c. Section 3 . 1 , problems * 2 , 4 , * 8 , * 10 . *2. Suppose A is a 3 × 3 matrix with the property that, for every other 3 × 3 matrix X , we have AX = XA . Show that A = r 0 0 0 r 0 0 0 r for some real number r . (Hint: if it commutes with every X , then it commutes with any matrix you choose; choose simple ones, such as the elementary matrices we dis- cussed in class.) 3. Show that the result in the previous problem is also true for n × n matrices, no matter what n is.
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