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Unformatted text preview: AB = BA + [ A, B ]. (a) Compute [ A, B ] for A = h 1 1 i and B = h 1 2 i . (b) For the same matrices A and B , compute [ A, [ A, B ]]. (c) Using only the associative property of matrix multiplication, show that for any three n n matrices A , B and C , [ A, [ B, C ]] + [ B, [ C, A ]] + [ C, [ A, B ]] = 0 . This is known as Jacobis identity . (We wont make use of it later. For our purposes, checking it is simply a good exercise in working with matrices.) SOLUTION (a) [ A, B ] = 11 . (b) Let C = [ A, B ]. From part (a) you know that C = 11 Now you just compute the commutator the way you did previously: [ A, [ A, B ]] = ACCA = 2 2 For (a) , we work out, using the denitions that we just practiced with: [ A, [ B, C ]] = ABC + CBAACBBCA 21 /september/ 2005; 22:09 21...
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 Fall '08
 Gladue
 Calculus, Matrices, Square Roots, Sets

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