Adv Alegbra HW Solutions 136

Adv Alegbra HW Solutions 136 - 136 (viii) If A = 1 2 , then...

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136 (viii) If A = £ 12 34 ± , then A 2 5 A 2 I = 0. Solution. True. (ix) Every n × n matrix over R has a real eigenvalue. Solution. False. (x) h 217 018 000 i is diagonalizable. Solution. True. 4.47 Let R be a commutative ring, let D : Mat n ( R ) R be a determinant function, and let A be an n × n matrix with rows α 1 ,...,α n .D e f ne d i : R n R by d i (β) = D i i 1 ,β,α i + 1 n ) . (i) If i ±= j and r R , prove that d i ( r α j ) = 0 . Solution. Absent. (ii) If i ±= j and r R , prove that d i i + r α j ) = D ( A ) . Solution. Absent. (iii) If r j R , prove that d i i + X j ±= i r j α j ) = D ( A ). Solution. Absent. 4.48 If O is an orthogonal matrix, prove that det ( O ) 1. Solution. Absent. 4.49 If A 0 is obtained from an n × n matrix by interchanging two of its rows, prove that det ( A 0 ) =− det ( A ) . Solution. Absent. 4.50 If A is an n × n matrix over a commutative ring R and if r R , prove that det ( rA ) = r n det ( A ) . In particular, det
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This note was uploaded on 12/21/2011 for the course MAS 4301 taught by Professor Keithcornell during the Fall '11 term at UNF.

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