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# hmw8 - Solutions to Homework 8 Math 110 Fall 2006 Prob...

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Solutions to Homework 8. Math 110, Fall 2006. Prob 4.1.1. (a) False, it is 2-linear. (b) True. (c) False, A is invertible if and only if det( A ) = 0. (d) False, it is the absolute value of that determinant. (e) True (proved in this section). Prob 4.1.2. (a) 30. (b) -17. (c) -8. Prob 4.1.4. For each pair of vectors, we need to find the absolute value of the determinant composed of the coordinates of these vectors. (a) 19. (b) 10. (c) 14. (d) 26. Prob 4.1.10. (a) By direct computation, CA = A 22 - A 12 - A 21 A 11 · A 11 A 12 A 21 A 2 = A 11 A 22 - A 12 A 21 0 0 A 11 A 22 - A 12 A 21 = A 11 A 12 A 21 A 22 · A 22 - A 12 - A 21 A 11 = AC = det( A ) I. (b) det( C ) = A 11 A 22 - A 12 A 22 = det( A ). (c) If A 12 and A 21 are interchanged in the matrix A , then they are also interchanged in the matrix C , i.e., the classical adjoint of A t is C t . (d) If A is invertible, then det( A ) = 0, so using (a), we obtain A (det( A ) - 1 C ) = (det( A ) - 1 C ) A = I , i.e., A - 1 = det( A ) - 1 C . Prob 4.2.1. (a) False, it is a multilinear transformation. (b) True. (c) True. (d) True. (e) False, the determinant gets multiplied by the scalar as well. (f) False, the determinant does not change. (g) False, if A has full rank, then it is invertible, hence its determinant is nonzero. (h) True.

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hmw8 - Solutions to Homework 8 Math 110 Fall 2006 Prob...

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