MATH 2310 HW1 Spring '10 - A , B and C are 4 4 matrices...

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Math 2310 Prelim-1 March 10, 2010 Instructions. This is a take home prelim. Solutions are due on Monday, March 15 at the beginning of class. Late submissions shall NOT be accepted under any circumstances. You should work individually. The only person you may ask doubts regarding the required material in this time frame is the course instructor. This prelim consists of two parts. Part I comprises routine prob- lems. Part II has problems that are not so routine. Part I Given that this is a take home test, I expect careful work here. Careless computational mistakes in this part shall be punished severely. Show your work clearly. 1.(i) Find the inverse of the matrix A = 1 1 1 1 1 3 1 2 1 2 - 1 1 5 9 1 6 . (ii) Express A as a product of elementary matrices. (5+5=10 points) 1
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2. Let A be the matrix ± 5 4 2 3 . (i) Find a vector x such that A x = 7 x . (ii) For which values of c does there exist a nonzero vector y satisfying A y = c y ? (5+5=10 points) 3. (i) Evaluate det( A ) where A is the matrix in question 1. (ii) Suppose that
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Unformatted text preview: A , B and C are 4 4 matrices with C = ( B T ) 2 . Further, given that det( A ) = 2 and det( B ) = 3. Find det(3 A-3 B 2 C-2 ). (5+5=10 points) Part II Solutions to problems here MUST BE WRITTEN IN DETAIL. You need not reprove theorems that I stated as such. However, if you use something from a problem that you have worked out before, you need to show those steps as well. 4. Do there exist invertible 3 3 matrices X and Y such that XY X-1 Y-1 = 1 0 2 0 2 1 0 0 1 ? Justify your answer rigorously. Hint: Think whether I have done a problem in class that is ideologically similar. ... (10 points) 5. Let A be an n n matrix. Show that the coecient of x n-1 in det( xI-A ) is equal to-Tr( A ). (10 points). 6. Find 2010 pairwise distinct 2 2 matrices A such that A 2010 = 1 0 0 1 . Hint: The key to this could lie with some geometry that we discussed. ...(10 points) 2...
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MATH 2310 HW1 Spring '10 - A , B and C are 4 4 matrices...

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