1111_08_t1sol - MATH1111/29Sept08/Test1 1 THE UNIVERSITY OF...

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MATH1111/29Sept08/Test1 1 THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS Linear Algebra Test 1 Solution Outline 1. (12 marks) Consider a linear system whose augmented matrix is of the form 1 2 1 0 2 5 3 0 - 1 1 β 0 . (a) Is it possible for the system to be inconsistent? Explain. (b) For what values of β will the system have infinitely many solutions? Ans . (a) (5 marks) No. The system always has the trivial solution (i.e. all variables equal zero). So it is always consistent. (b) (7 marks) By applying elementary row operations, we have 1 2 1 0 2 5 3 0 - 1 1 β 0 - 2 R 1 + R 2 -----------→ 1 2 1 0 0 1 1 0 - 1 1 β 0 R 1 + R 3 -----------→ 1 2 1 0 0 1 1 0 0 3 β + 1 0 - 3 R 2 + R 3 -----------→ 1 2 1 0 0 1 1 0 0 0 β - 2 0 . The consistent system has free variable if and only if β - 2 = 0. So the system has infinitely many solution if and only if β = 2. 2. (12 marks) (a) Reduce the following matrix A into reduced row echelon form. A = 2 1 - 3 6 0 0 - 1 2 0 1 - 1 4 - 1 - 1 1 2 .
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