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 ...”]. Can
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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 flaw 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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