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 Jacobi’s identity . (We won’t make use of it later. For our purposes, checking it is simply a good exercise in working with matrices.) SOLUTION (a) [ A, B ] = ± 11 ² . (b) Let C = [ A, B ]. From part (a) you know that C = ± 11 ² Now you just compute the commutator the way you did previously: [ A, [ A, B ]] = ACCA = ±2 2 ² For (a) , we work out, using the deﬁnitions that we just practiced with: [ A, [ B, C ]] = ABC + CBAACBBCA 21 /september/ 2005; 22:09 21...
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This note was uploaded on 10/03/2010 for the course MATH 380 taught by Professor Gladue during the Fall '08 term at Roger Williams.
 Fall '08
 Gladue
 Calculus, Matrices, Square Roots, Sets

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