Science:Math Exam Resources/Courses/MATH152/April 2011/Question A 04

MATH152 April 2011

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Question A 04
Circle the one correct answer below. A linear system of two equations in three unknowns has (a) always a unique solution. (b) either a unique solution or no solutions. (c) either a unique solution or an infinite number of solutions. (d) either no solutions or an infinite number of solutions. (e) either a unique solution or exactly two distinct 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
Consider how many pivots and free variables there are with 3 unknowns and two equations.

Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. With two equations (rows in the matrix), there can be at most 2 pivots and thus there is at least one free variable. A free variable implies that there are an infinite number of solutions provided that the right hand side vector vanishes wherever there is a row of zeros, otherwise a solution does not exist. Therefore, we either have at least one valid free variable which implies that there are an infinite number of solutions or the right hand side vector does not vanish whenever there is a row of zeros. Therefore we either have no solutions or an infinite number of solutions when a linear system has 2 equations with three unknowns. Therefore, the correct option is (d). Advanced: Recall that the rank (r) of a matrix defines the number of linearly independent rows or columns of a matrix or linear system and recall that for any matrix A size m by n that, $r\leq \min(n,m).$ For this example we have m=2 and n=3 and so r ≤ 2. The dimension of the nullspace is n-r and in this case it is at least 1 which implies that there are non-trivial solutions to Ax = 0 and in fact there are an infinite number of them. A solution to Ax = b exists as long as b is spanned by the column space of A. If this is true then Ax = b has a particular solution along with the infinite set of solutions belonging to the nullspace and therefore overall there are an infinite number of solutions. However, if b is not spanned by the column space of A then there are no solutions to Ax = b. Therefore, we conclude that a linear system with 2 equations in 3 unknowns has (d) either no solutions or an infinite number of solutions.