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# HW2-250A,B - HOMEWORK 2(Math 250 A B a b 1(a Given A c d...

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HOMEWORK 2 (Math 250 A, B) 1. (a) Given d c b a A , show that: (20 pts) (i) 2 ) ( x x p A Tr( A ) x + det( A ) (ii) The eigenvalues of A are 2 4 ) ( ) ( 2 2 , 1 bc d a d a . (b) Let A be an n x n invertible matrix. Then, show that: (i) All the eigenvalues of A are non-zero. (ii) The eigenvalues of 1 A are of the form 1 , where is an eigenvalue of A . (iii) ) / 1 ( ) det( ) ( ) ( 1 x p A x x p A n A . ( Hint : For (b) part (iii) , use the fact that det( AB ) = det( A )det( B ) ) 2. (a) If A is an n x n diagonalizable matrix, then show that P D P A k k 1 , where k , D is diagonal, and P is some invertible matrix . (20 pts) (b) Use part (a) to find 11 A , where 2 15 0 0 1 0 1 7 1 A . ( Hint : For (a), use the fact that since A is diagonalizable then A is similar to a diagonal matrix D ) 3. Show that A and B are orthogonal vectors in an inner product space V if and only if 2 2 2 B A X where B A X . The above, is usually called the

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HW2-250A,B - HOMEWORK 2(Math 250 A B a b 1(a Given A c d...

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