Linear Algebra with Applications (3rd Edition)

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110.201 Linear Algebra 5th Quiz April 21, 2005 Problem 1 Find the determinant of the n × n matrix A = 0 0 ··· 0 1 0 0 ··· 1 0 . . . . . . . . . . . . . . . 0 1 ··· 0 0 1 0 ··· 0 0 . Problem 2 Let A be an n × n matrix obeying the equation A 2 = A. a) What are the possible values of det( A )? Why? b) Let V be the image of A and m = dim V Find all relationships between m,n and the values for det( A ) you found above [An acceptable statement would be something like “If det( A ) = ... , then
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Unformatted text preview: ...]. Can m = n ? If so, and m = n , what can you say about A ? Problem 3 Suppose that two square matrices satisfy the following identity AB =-BA . Find the aw in the following argument, showing a counterexam-ple: Taking determinants gives (det A )(det B ) =-(det B )(det A ), so either A or B must have zero determinant. Thus AB =-BA is only possible if A or B is singular. 1...
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This homework help was uploaded on 01/23/2008 for the course MATH 201 taught by Professor Consani during the Spring '05 term at Johns Hopkins.

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