23_hwc1SolnsODDA

# 23_hwc1SolnsODDA - A = 3 2-3-2 , and you can easily check...

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Looking at the oof diagonal entries we see that either x = 0 or y = 0. But if y = 0, then the diagonal entries would be x 2 , which cannot equal - 4. Hence it must be that x = 0 and y 2 = 4, or y = ± 2. 3.15 Find all 2 × 2 matrices A such that A 2 = A . SOLUTION Let A = ± a b c d ² . Then A 2 = ± a 2 + bc ( a + d ) b ( a + d ) c d 2 + bc ² . Hence A 2 = A if and only if each of the following equations are satisﬁed: a = a 2 + bc d = d 2 + bc b = ( a + d ) b and c = ( a + d ) c . From here, it is just a matter of being very systematic: case 1 Suppose that b = c = 0. Then the equations reduce to a = a 2 and d = d 2 , so both a and d are equal to either 0 or 1. This gives us the four matrices ± 1 0 0 1 ² ± 1 0 0 0 ² ± 0 0 0 1 ² and ± 0 0 0 0 ² . case 2 Suppose that b 6 = 0. Then from b = ( a + d ) b we have that a + d = 1. Then c = ( a + d ) c is automatically satisﬁed, and using d = 1 - a , d = d 2 + bc is the same as a = a 2 + bc . Hence all of the equations will be satisﬁed if c = ( a - a 2 ) /b . This gives us the matrices ± a b ( a - a 2 ) /b 1 - a ² , where a is any number and b is any non zero number. For example, with a = 3 and b = 2, this gives us
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Unformatted text preview: A = 3 2-3-2 , and you can easily check that A 2 = A . case 3 Suppose that c 6 = 0. (Note that there is overlap between cases 2 and 3.) Then, proceeding as in case 2, we are led to the matrices a ( a-a 2 ) /c c 1-a , where a is any number and c is any non zero number. 3.17 Does there exist an invertible 2 2 matrix A such that A 2 = 0. If so, nd an example. Otherwise, explain why not. SOLUTION No. Suppose that A is invertible, and let B be the inverse of A , so that BA = I . Then B 2 A 2 = BBAA = B ( BA ) A = BA = I . But if A 2 = 0, B 2 A 2 = 0. Since B 2 A 2 cannot equal both I and 0, there is no invertible matrix A with A 2 = 0. 21 /september/ 2005; 22:09 24...
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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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