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mat1302-practice-midterm1-2008-solutions

# mat1302-practice-midterm1-2008-solutions - MATH 1302...

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MATH 1302 - PRACTICE FIRST MIDTERM EXAM SOLUTIONS WINTER 2008 1. a) If a linear system consists of 4 equations and 5 unknowns, what is the maximum possible number of pivot positions in the corresponding coefficient matrix? b) If a linear system consists of 10 equations and 7 unknowns, what is the maximum possible number of pivot positions in the corresponding augmented matrix? c) If you know the number of pivot positions in both the coefficient matrix and the augmented matrix corresponding to a linear system, how can you tell if the linear system is consistent or not? d) Does every matrix have a unique (i.e. only one) reduced echelon form? e) Which of the following matrices are in echelon form? A = 1 0 0 0 1 1 0 0 0 0 1 0 , B = - 1 0 0 0 4 0 0 0 π , C = 3 - 4 0 7 0 8 10 - 11 0 0 0 18 0 0 0 0 Answer: a) The maximum possible number of pivot positions is 4. b) The maximum possible number of pivot positions is 8. c) The linear system is consistent if and only if the coefficient and augmented matrices have the same number of pivot positions. d) Yes. e) The matrices B and C are in echelon form. The matrix A is not. 2. Let u = 1 - 2 , v - a 8 , w = b 7 . For which values of a and b does the vector equation x u + y v = w have a) no solution, b) exactly one solution, c) infinitely many solutions?

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2 MAT 1302 - PRACTICE FIRST MIDTERM EXAM SOLUTIONS Answer: The given vector equation has the same solution set as the linear system whose augmented matrix is £ u v w / = 1 - a - 2 8 fl fl fl fl b 7 R 2 +2 R
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