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Consider the systems, each consisting of m linear equations in n variables. I. If m < n, then all such systems have a solution II. If m > n, then none of these systems has a solution III. If m = n, then there exists a system which has a solution Which one of the following is CORRECT?

1. I, II and III are true
2. Only II and III are true
3. Only III is true
4. None of them is true

Correct answer: Only III is true

Solution

Statement III is correct because when the number of equations equals the number of variables, it is possible to have a consistent system that has a solution. Statement I is false as having fewer equations than variables does not guarantee a solution, and Statement II is also false since it is possible for an overdetermined system to still have a solution.

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