fa11_mae260_hw4 - MAE 260 Homework Assignment#4 Issued Nov...

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MAE 260. Homework Assignment #4 Issued: Nov. 10, 2011 Due: Nov. 24, 2011 (in class) 1. The trace of a square matrix A R n × n , denoted as Tr ( A ) stands for the sum of all the diagonal elements: Tr ( A ) = n i = 1 A ii . Show that: (a) Tr ( AB ) = Tr ( BA ) . (b) Tr ( ABC ) = Tr ( BCA ) = Tr ( CAB ) . 2. Come up with matrix A such that (a) A 0 but A 2 = 0 . (b) A 2 0 but A 3 = 0 . 3. Suppose you solve A x = b for three special right sides b : A x 1 = 1 0 0 , and A x 2 = 0 1 0 , and A x 3 = 0 0 1 4. Find all matrices A = a b c d that satisfy A 1 1 1 1 = 1 1 1 1 A . 5. Show that matrices A and [ A AB ] (with extra columns) have the same column space. (Hint: you need to show that every column of
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