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HW5_Solution - 1 3 1 2 4 6 A 2 6 2 B 1 3 5 1 3 1 3 5 8 1(i...

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EEL3105 Homework 5 Solutions 1. 1 3 1 2 6 2 1 3 1 A 2 4 6 1 3 5 3 5 8 B (i) 2 8 13 4 16 26 2 8 13 AB 4 12 4 2 6 2 5 15 5 BA 2 4 17 2 10 28 7 23 8 AB BA ( ) 5 ( ) 5 ( ) 0 trace AB trace BA trace AB BA (ii) 11 12 1 21 22 2 1 2 .. .. .. .. . . . . . . . . . . . . n n n n nn a a a a a a A a a a 11 12 1 21 22 2 1 2 .. .. .. .. . . . . . . . . . . . . n n n n nn b b b b b b B b b b 1 1 1 2 2 1 1 .. .. .. .. . .. .. .. . . . . . . . . . . .. .. . . n j j j n j j j n nj jn j a b a b AB a b 1 1 1 2 2 1 1 .. .. .. .. . .. .. .. . . . . . . . . . . .. .. . . n j j j n j j j n nj jn j b a b a BA b a
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1 1 1 1 1 1 2 2 2 2 1 1 1 1 .. .. .. .. . .. .. .. . . . . . . . . . . .. .. . . n n j j j j j j n n j j j j j j n n nj jn nj jn j j a b b a a b b a AB BA a b b a 1 1 1 1 2 2 2 2 1 1 1 1 1 1 ( ) ......... 0 n n n n n n j j j j j j j j nj jn nj jn j j j j j j trace AB BA a b b a a b b a a b b a (iii) 11 12 1 21 22 2 1 2 .. .. .. .. . . . . . . . . . . . . n n m m mn a a a a a a A a a a (A, mxn matrix) 11 12 1 21 22 2 1 1 .. ..
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