20_hwc1SolnsODDA

# 20_hwc1SolnsODDA - AB = BA A B(a Compute A B for A = h 1 1...

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(b) Find all possible values of a b and c so that A 2 = B where B = h 1 3 0 4 i . How many diﬀerent sets of values for a b and c are there for which A 2 = B ? (The matrices A that you compute here are square roots of the matrix B .) SOLUTION (a) ± a 2 ab + bc 0 c 2 ² . (b) We see by comparing the upper left entries that we have to have a 2 = 1, so we have to have a = ± 1. We see by comparing the lower right entries that we have to have c 2 = 4, so we have to have c = ± 2. This gives us a total of four choices for the values of a and c . But then, whichever of these four we choose, we can go on to choose exactly one value of b so that we get equality in the upper right entries. (The lower left entries are always zero, and so they take care of themselves.) Thus, there are four diﬀerent “sqaure roots” of the matrix B : ± 1 1 0 2 ² ± 1 - 3 0 - 2 ² ± - 1 1 0 2 ² and ± - 1 - 1 0 - 2 ² . 3.9 Given two n × n matrices A and B , the commutator of A and B is deﬁned to be the n × n matrix AB - BA , and is denoted by [ A, B ]. The reason for this is that AB = BA if and only if [ A, B ] = 0. More generally, but also directly from the deﬁnition,
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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 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 ] = ± 1-1 ² . (b) Let C = [ A, B ]. From part (a) you know that C = ± 1-1 ² Now you just compute the commutator the way you did previously: [ A, [ A, B ]] = AC-CA = ±-2 2 ² For (a) , we work out, using the deﬁnitions that we just practiced with: [ A, [ B, C ]] = ABC + CBA-ACB-BCA 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.

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