winter 2007assign2

winter 2007assign2 - An equivalent statement for the...

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Math 235 Assignment 2 Due 9:15am, Wednesday Jan 24, 2007. 1. From the Text, § 3.1 #14. § 3.2 #10, #22. § 3 (Supplementary Exercises, page 211+) #14, #16. 2. Let A = [ a ij ] be an n × n matrix (over C , say). Let A ij denote the submatrix obtained from A by removing the i -th row and the j -th column, and let C ij be the cofactors of A defined by C ij = ( - 1) i + j det ( A ij ) . The matrix formed by the cofactors adj ( A ) = C 11 C 21 · · · C n 1 C 12 C 22 · · · C n 2 · · · · · · · · · · · · C 1 n C 2 n · · · C nn is called the classical adjoint of A . The fact that ( A )(adj ( A )) = det ( A ) I n is a theorem found in many text books. (cf. Theorem 8, page 203).
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Unformatted text preview: An equivalent statement for the theorem is that n j =1 a ij C kj = 0 for all i = k, (0.1) and n j =1 a ij C kj = det ( A ) for all i = k. (0.2) The latter is just the cofactor expansion for A along the i-th row. Give support for (0.1): For xed given i = k , exhibit a matrix B having two equal rows (or columns, thus det ( B ) = 0 is seen easily ) such that n j =1 a ij C kj (0.3) can be recognized as a cofactor expansion of B ....
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This note was uploaded on 05/23/2010 for the course MATH 235 taught by Professor Celmin during the Winter '08 term at Waterloo.

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