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HWSolutions1.2 - Newberger Math 247 Spring 03 Homework...

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Newberger Math 247 Spring 03 Homework solutions: Section 1.2, #23-26, 29, 30 23. Suppose that a 3 × 5 coefficient matrix for a system has three pivot columns. Is the system consistent? This system is consistent. The coefficient matrix has 3 rows. Since the coefficient matrix has 3 pivots, there is a pivot in each row. This means each row in the coefficient matrix has a non-zero entry. So in the augmented matrix there cannot be a row of the form £ 0 0 0 0 0 b / where b 6 = 0 . By Theorem 2, the system is consistent. 24. Suppose that a system of linear equations has a 3 × 5 augmented matrix whose fifth column is a pivot column. Is the system consistent? This system is inconsistent. The fifth column of the augmented matix is the rightmost column and since it is a pivot column, the matrix has a row of the form £ 0 0 0 0 b / where b 6 = 0 . By Theorem 2, the system is inconsistent. 25. Suppose the coefficient matrix of a system of linear equations has a pivot position in every row. Explain why the system is consistent.
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