asst2 - c.) For each eigenvalue of A , nd the associated...

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MATH 235, Fall 2007 Assignment #2 1. From the text: Section 3.3 # 6, 14 2. Let A Z n × n ,b Z n , and suppose that det ( A ) = 1. Prove that the system Ax = b has exactly one solution x = ( x 1 ,...,x n ), and that x 1 ,...,x n Z . (Note: A Z n × n means that A is an n × n matrix with entries from Z ). 3. Let A R n × n ,b R n . Prove that det ( A i ( b )) = ( adj ( A ) b ) i for all i ∈ { 1 ,...,n } , where adj ( A ) denotes the classical adjoint (or adjugate) of A. (Note: If x R n and i ∈ { 1 ,...,n } , then x i denotes the i th entry in x ). 4. From the text: Section 3.3 # 24 5. Prove that the equation of the line in R 2 which passes through the distinct points ( x 1 ,y 1 ) and ( x 2 ,y 2 ) can be written as 0 = det x y 1 x 1 y 1 1 x 2 y 2 1 6. Prove that the following four points are coplanar in R 3 : ( - 6 , 0 , 0) , (12 , 12 , 0) , (2 , 0 , 4) , (0 , - 8 , 9) . 7. From the text: Section 5.1 # 16, 26 8. For each of the following matrices A (( i ) and ( ii )): a.) Find the characteristic polynomial of A . b.) Find all the eigenvalues of A , and state each of their multiplicities.
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Unformatted text preview: c.) For each eigenvalue of A , nd the associated eigenspace E and express it as a span of basis vectors for that eigenspace. ( i ) A = 7 10-2-5 ( ii ) A = 0 2 1 3-8 0 4 0 6 4 0 0 4 2-4 0 0 0 0 0 0 0 0-1 9. From the text: Section 5.2 # 20 10. Prove that for any positive integer n , the n n matrix A dened by A = -c 1 -c 1 1 . . . . . . . . . . . . . . . . . .-c n-2 1-c n-1 has characteristic equation (-1) n ( n + c n-1 n-1 + ... + c 1 + c ) = 0 . (Note: If n = 1 then A = [-c ]; if n = 2 then A = -c 1-c 1 ; etc.)....
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This note was uploaded on 04/27/2010 for the course MATH 235 taught by Professor Celmin during the Fall '08 term at Waterloo.

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