08aAss1sol

08aAss1sol - AS1sol/MATH1111/YKL/08-09 THE UNIVERSITY OF...

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Unformatted text preview: AS1sol/MATH1111/YKL/08-09 THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS MATH1111: Linear Algebra Assignment 1 Suggested Solution 1. Substituting x = 3, y =- 1, z = 2 in turn into the equations, we obtain a system of linear equations in unknowns a,b,c :    3- a + 2 c = 0 3 b- c- 6 = 1 3 a- 2 + 2 b = 5 Its augmented matrix is  - 1 0 2- 3 3- 1 7 3 2 7   3 R 1 + R 3-----------→  - 1 0 2- 3 3- 1 7 2 6- 2  - 2 3 R 2 + R 3-----------→  - 1 0 2- 3 3- 1 7 20 3- 20 3  - R 1 , 1 3 R 2 , 3 20 R 3-----------→   1 0- 2 3 0 1- 1 3 7 3 0 0 1- 1   . Hence the solution is a = 1 , b = 2 , c =- 1. 2. ”if part”: Let AB be symmetric, i.e. ( AB ) T = AB . The L.H.S. is ( AB ) T = B T A T = BA, as A and B are symmetric. So AB = BA . ”only if part”: Assume AB = BA . Consider the transpose of AB , ( AB ) T = B T A T = BA as A and B are symmetric. This follows ( AB ) T = BA = AB by our condition. Hence AB is symmetric. 3. Let A be a square matrix of order n . Prove the following: (a) As A is invertible, its inverse A- 1 exists. Thus, from AB = 0, we have...
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This note was uploaded on 09/06/2010 for the course MATH MATH1111 taught by Professor Forgot during the Fall '08 term at HKU.

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08aAss1sol - AS1sol/MATH1111/YKL/08-09 THE UNIVERSITY OF...

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