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Unformatted text preview: n such that there exists an n by n matrix A with coecients in Q satisfying A 3 = 2 I n . (Here I n denotes the n by n identity matrix.) 5. (14 points) Consider the elementary Jordan matrix A = 1 1 0 of size n by n over a eld K . Determine the Jordan canonical form of A 2 . 6. (12 points) Let R be a domain and A an n by n matrix with entries in R , where n 2. Prove that det(adj( A )) = (det( A )) n-1 . (Recall that adj( A ) is the n by n matrix whose ( i,j )-entry is (-1) i + j times the determinant of the matrix obtained from A by deleting row j and column i .) 7. (15 points) Let V be a nite-dimensional vector space over C . Show that f is a symmetric bilinear form of rank at most 2 on V if and only if there exist and in V * such that f ( x,y ) = ( x ) ( y ) + ( y ) ( x ) for every x and y in V ....
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This document was uploaded on 01/25/2012.
- Spring '09