Discussion
Consider the systems, each consisting of \(m\) linear equations in \(n\) variables.
I. If \(m \lt n\), then all such systems have a solution
II. If \(m \gt n\), 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?
Answer: C
Only III is true.
Statement I is false: a system with fewer equations than unknowns can still be inconsistent, for example \(x + y + z = 1\) and \(x + y + z = 2\) (\(m = 2\), \(n = 3\)).
Statement II is false: more equations than unknowns can still be consistent, for example \(x = 1\), \(2x = 2\), \(3x = 3\) (\(m = 3\), \(n = 1\)); any homogeneous system also has the solution 0.
Statement III is true: it only claims that some square system has a solution, and \(x = 1\) (or any system with an invertible coefficient matrix) is one.
So options including I or II are wrong, and “none of them” ignores III.
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