Unformatted text preview: n such that there exists an n by n matrix A with coeﬃcients 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 ﬁeld 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 )) n1 . (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 ﬁnitedimensional 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 ϕ and ψ in V * such that f ( x,y ) = ϕ ( x ) ψ ( y ) + ϕ ( y ) ψ ( x ) for every x and y in V ....
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 Spring '09
 Linear Algebra, Vector Space, Ring, Abelian group

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