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Unformatted text preview: R is a ﬁeld and n ≤ m , show that the characteristic polynomials p AB ( x ) and p BA ( x ) of AB and BA respectively satisfy p AB ( x ) = x mn p BA ( x ). 6. Let A = [ a ij ] be the ( n + 1) × ( n + 1) matrix with a ij = ( i + j2)! and 0! = 1. (Hint: A = LDL T ) (a) A is positive deﬁnite. (b) det A = (0!1! ··· n !) 2 . (c) ( n !) 2 A1 ∈ M ( n +1)( n +1) ( Z ). 1 7. Let A be an n × n matrix over C and p be a prime number. Suppose that I 6 = A and A p = I and tr( A ) = positive integer ‘ . Show that n = ‘ + sp with s a positive integer. 2...
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 Spring '09
 Scalar, Vector Space, Ring, Prime number

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