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# ps1 -    x 1-2 x 3 x 4 x 5 = 0 2 x 1-x 2 x 3-3 x...

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University of Hong Kong Fall 2011 Math 1813 Mathematical Methods for Actuarial Science Instructor: Siye Wu Problem Set 1 (due Thursday, 22 September) I. Prove or disprove the following. (a) Suppose that A,B M n,n ( R ) are two matrices satisfying AB + BA = 0. (i) Either A or B is singular in the case that n is odd; (ii) Either A or B is singular in the case that n is even. (b) Let A M m,n ( R ). Suppose that A T A x = 0 for some nonzero x M n, 1 ( R ). (i) det( A T A ) = 0; (ii) det( AA T ) = 0; (iii) If m = n , then det( A ) = 0. II. Suppose A M m × n ( R ) and B M n × m ( R ). (a) Show that the matrix ± I m + AB A B I n ² is invertible and its inverse is ± I m - A - B I n + BA ² . (b) If the above inverse is not given to you, ﬁnd it by using the identity ± I m + AB A B I n ² = ± I m A O I n ²± I m O B I n ² . III. Find the determinants of the following matrices. (a) F 4 = 2 - 1 0 0 - 1 2 - 2 0 0 - 1 2 - 1 0 0 - 1 2 . (b) a 2 ( a + 1) 2 ( a + 2) 2 ( a + 3) 2 b 2 ( b + 1) 2 ( b + 2) 2 ( b + 3) 2 c 2 ( c + 1) 2 ( c + 2) 2 ( c + 3) 2 d 2 ( d + 1) 2 ( d + 2) 2 ( d + 3) 2 . IV. (a) Find the solution(s), if any, to the system of linear equations
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Unformatted text preview:    x 1-2 x 3 + x 4 + x 5 = 0 2 x 1-x 2 + x 3-3 x 4-x 5 = 0 9 x 1-3 x 2-x 3-7 x 4 = 4 by using the Gauss-Jordan elimination. (b) Find the inverse of F 4 in III(a) by transforming ( F 4 | I 4 ) into its reduced row echelon form. V. (a) Find the condition on the real numbers α,β,γ such that the linear system    x + αy + α 2 z = α 3 x + βy + β 2 z = β 3 x + γy + γ 2 z = γ 3 has a unique solution, and ﬁnd the unique solution under such a condition using Cramer’s rule. (b) Discuss the solutions of the above linear system when the condition for the uniqueness of solution is not satisﬁed....
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