GATE CS 2016 Set 2 — Question 14
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Engineering Mathematics → Linear Algebra → System of Linear Equations
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Question
Consider the systems, each consisting of linear equations in variables.I. If , then all such systems have a solution
II. If , then none of these systems has a solution
III. If , then there exists a system which has a solutionWhich one of the following is CORRECT?
II. If , then none of these systems has a solution
III. If , then there exists a system which has a solutionWhich one of the following is CORRECT?
Correct answer
(C) Only III is true
Solution
Let the system be , where is an matrix.Statement I: If (fewer equations than variables), the system is underdetermined. While such systems often have infinite solutions if consistent, they can also be inconsistent (no solution). For example, and has but no solution. Thus, statement I is false.Statement II: If (more equations than variables), the system is overdetermined. While such systems are often inconsistent, they can have a solution if the equations are not independent and are consistent. For example, and has and a unique solution . Thus, statement II is false.Statement III: If , there exists a system which has a solution. This is true. For example, the system (homogeneous system with identity matrix) always has the trivial solution . Thus, statement III is true.Therefore, only statement III is correct.
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