Science:Math Exam Resources/Courses/MATH152/April 2012/Question 06 (d)
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Question 06 (d) 

This question requires little or no calculations. Consider a linear system of 3 equations with 5 unknowns. Answer True or False to each of the following statements (no justification required). (1) It always has at least one solution. (2) There is either no solution or infinitely many solutions. (3) If a solution exists, then there is precisely a 2parameter family of solutions. (4) If the coefficient matrix has rank k, then the associated homogeneous system as a kparameter family of solutions. 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

Try to determine a counterexample first. If you find it difficult to determine one, the answer may be true. 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. (1) It always has at least one solution. (FALSE) (2) There is either no solution or infinitely many solutions. (TRUE) (3) If a solution exists, then there is precisely a 2parameter family of solutions. (FALSE) (4) If the coefficient matrix has rank k, then the associated homogeneous system as a kparameter family of solutions. (FALSE) The justification for each answer is given below. (1) is FALSE. Consider the following counterexample of three equations with five unknowns: Clearly, this system is inconsistent. Hence (1) is FALSE by counterexample. (2) is TRUE. As we saw in (1), it is possible that there is no solution. For the second part of the statement, assume that a solution exists. This solution cannot be unique since any rowreduced echelon form of the coefficient matrix admitting a solution has at least two free parameters (i.e. Rank of the matrix is at most 3, but there are 5 parameters). Hence, if any solution exists, then infinitely many solutions must exist. Therefore (2) is true. (3) is FALSE. Consider the following example: In this example, there is a fourparameter family of solutions since the associated coefficient matrix has rank 1. (4) is FALSE. See (1) and (3) for counterexamples. 