tut01s - THE UNIVERSITY OF SYDNEY PURE MATHEMATICS Linear...

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Unformatted text preview: THE UNIVERSITY OF SYDNEY PURE MATHEMATICS Linear Mathematics 2012 Tutorial 1 Solutions 1. a ) Calculate the matrix product 3 0 4 1 1 2 1 3 5 5 2 4 . b ) Hence find the solution to the system of linear equations that corresponds to the augmented matrix 3 0 4 31 1 1 2 11 1 3 5 9 . Solution a ) 3 0 4 1 1 2 1 3 5 5 2 4 = 31 11 9 . b ) Let A = parenleftBig 3 0 4 1 1 2- 1 3 5 parenrightBig . Then we want to find x = parenleftBig x 1 x 2 x 3 parenrightBig such that A x = parenleftBig 31 11 9 parenrightBig . From part (a) x = 5 2 4 is a solution. The question, then, is whether or not this is the only solution. One way to see that it is is to evaluate det A . We have det A = 13 negationslash = 0 (check!). Hence A is invertible, and the system has the unique solution x = 5 2 4 . We could, of course, find the solution using Gaussian elimination: 3 0 4 31 1 1 2 11 1 3 5 9 R 1 R 2 1 1 2 11 3 0 4 31 1 3 5 9 R 2 := R 2- 3 R 1 1 1 2 11 3 2 2 1 3 5 9 R 3 := R 3 + R 1 1 1 2 11 3 2 2 4 7 20 R 2 := R 2 + R 1 1 1 2 11 0 1 5 18 0 4 7 20 R 3 := R 3- 4 R 2 1 1 2 11 0 1 5 18 0 0 13 52 R 3 :=- 1 13 R 3 1 1 2 11 0 1 5 18 0 0 1 4 Therefore x 3 = 4 , x 2 = 18 5 x 3 = 2 , x 1 = 11 x 2 2 x 3 = 5 . Using back substitution the unique solution is parenleftBig 5- 2 4 parenrightBig , as before. 2. Each of the following matrices is the reduced row echelon form of an augmented matrix be- longing to a system of linear equations in the variables x i , ( i = 1 , 2 , . . . ) . (Both the systems represented here have infinitely many solutions why?) For each augmented matrix (i) determine the number of parameters needed to solve the system and (ii) express the solution of the system in parametric form. Math 2061: Tutorial 1 Solutions A.M. 5/1/2012 Linear Mathematics Tutorial 1 Solutions Page 2 a ) parenleftBigg 1 4 1 5 1 parenrightBigg b ) parenleftBigg 1 2 6 1 5 1 1 parenrightBigg Solution a ) i ) The variable x 3 is a parameter....
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This note was uploaded on 02/06/2012 for the course MATH 2061 taught by Professor Notsure during the Three '09 term at University of Sydney.

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tut01s - THE UNIVERSITY OF SYDNEY PURE MATHEMATICS Linear...

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