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**Unformatted text preview: **R is a eld 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 m-n p BA ( x ). 6. Let A = [ a ij ] be the ( n + 1) ( n + 1) matrix with a ij = ( i + j-2)! and 0! = 1. (Hint: A = LDL T ) (a) A is positive denite. (b) det A = (0!1! n !) 2 . (c) ( n !) 2 A-1 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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