21_hwc1SolnsODDA

21_hwc1SolnsODDA - solve the third equation for v . This...

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[ B, [ C, A ]] = BCA + ACB - BAC - CAB [ C, [ A, B ]] = CAB + BAC - CBA - ABC Notice that each of the six products ABC CBA BCA ACB CAB BAC shows up once with a plus sign and once with a minus sign, so that when we form the sum [ A, [ B, C ]] + [ B, [ C, A ]] + [ C, [ A, B ]], everything cancels out. 3.11 For any three numbers a , b and c , let A = ± 1 a b 0 1 c 0 0 1 ² . For any three numbers u , v and w , let B = ± 1 u v 0 1 w 0 0 1 ² . (a) Compute the product AB . (b) Show that A is always invertible, and ﬁnd the inverse. (Your answer will be a matrix that, like A , depends on a , b and c .) SOLUTION (a) AB = 1 a + u aw + b + v 0 1 c + w 0 0 1 (b) The inverse is a matrix B such that AB = I . Using (a) you will ﬁnd that AB = I if and only if the system of equations a + u = 0 c + w = 0 aw + v + b = 0 is satisﬁed. The ﬁrst two equations tell you u = - a , and w = - c . With this it is easy to
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Unformatted text preview: solve the third equation for v . This gives you the values of u , v and w for which B = A-1 , and hence A-1 = 1-a ac-b 1-c 1 . 3.13 Let J = h-1 1 i , and let I denote the 2 2 identity matrix h 1 1 i . (a) Show that J 2 =-I , and that the transformation of IR 2 induced by J is simply the counterclockwise rotation about the origin through an angle of / 2. (This means that-I has a square root, and that the square root is something quite natural. Indeed, two 90 degree turns do make a u-turn.) 21 /september/ 2005; 22:09 22...
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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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