solution2 - Homework assignment 2 p 21 Exercise 2. Let A =...

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Unformatted text preview: Homework assignment 2 p 21 Exercise 2. Let A = 1- 1 1 2 1 3 1 , B = 2- 2 1 3 4 4 Verify directly that A ( AB ) = A 2 B Solution: A 2 = 2- 1 1 5- 2 3 6- 3 4 , A 2 B = 7- 3 20- 4 25- 5 AB = 5- 1 8 10- 2 , A ( AB ) = 7- 3 20- 4 25- 5 Exercise 3. Find two different 2 2 matrices A such that A 2 = 0 but A 6 = 0 . Solution: Let A = a b c d be a 2 2 matrix. Then A 2 = a 2 + bc ( a + d ) b ( a + d ) c d 2 + bc . Now find a,b,c,d such that a 2 + bc = ( a + d ) b = ( a + d ) c = d 2 + bc = 0 . Note that if a + d = 0 and ad- bc = 0 , then all these equations are satisfied. For instance, put a =- d = 1 ,b =- c = 1 or put a = d = 0 ,b = 0 ,c = 1 . Then the matrices 1 1- 1- 1 , 0 1 0 0 are not but their squares are . Exercise 4. For the matrix A of Exercise 2 , find elementary matrices E 1 ,E 2 ,...,E k such that E k E 2 E 1 A = I 1 Solution: Let us first row-reduce A into the identity matrix: A- 2 I + II + 3 1- 1 1 2- 1 3 1 - 3 I + III + 3 1- 1 1 2- 1 3- 2 1 2 II + 3 1- 1 1 1- 1 2 3- 2 - 3 II + III + 3 1- 1 1 1- 1 2- 1 2 - 2 III + 3 1- 1 1 1- 1 2 1 - 1 2 III + II + 3 1- 1 1 1 1 - 1 III + I + 3 1- 1 0 1 1 1 II + I + 3 1 0 0 0 1 0 0 0 1 Applying the transformation on top of each arrow to the identity matrix, we obtain the elementary transformations we want. For example, E 1 is obtained by applying- 2 I + II to the identity matrix, therefore: E 1 = 1 0 0- 2 1 0 0 1 Analogously we obtain all the other matrices, the last one is E 8 = 1 1 0 0 1 0 0 0 1 Exercise 7. Let A and B be 2 2 matrices such that AB = I . Prove that BA = I ....
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This note was uploaded on 11/14/2010 for the course PHYCS 498 taught by Professor Aa during the Spring '10 term at University of Illinois, Urbana Champaign.

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solution2 - Homework assignment 2 p 21 Exercise 2. Let A =...

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