ch4 - Solutions to Homework 9 Math 110 Fall 2006 Prob...

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Solutions to Homework 9. Math 110, Fall 2006. Prob 4.3.10. Since det( AB ) = det( A ) det( B ) for any two square matrices of the same order, this implies, by induction, that (det M ) k = det( M k ) for all k IN. Since the determinant of the zero matrix is zero, this shows that, for a nilpotent matrix M , (det M ) k = 0, hence det M = 0. Prob 4.3.11. By Theorem 4.8, det M t = det M . Now, if M is skew-symmetric, then det M = det M t = det( - M ) = ( - 1) n det M, the last equality by the n -linearity of the determinant. If n is odd, this gives det M = - det M , i.e., det M = 0. If n is even, it does not imply anything. Prob 4.3.14. Since β consists of n vectors, β is a basis if and only if these vectors are linearly independent, which is equivalent to the map L B being one-to-one. Since the matrix B is square, this is in turn equivalent to B being invertible, hence having a nonzero determinant. Thus, β is a basis if and only if det B = 0. Prob 4.3.15. If matrices A and B are similar, then, for some invertible matrix Q , we have A = QBQ - 1 .
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