Consider the systems, each consisting of m linear equations in n variables. I.…
2016
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?
- A.
I, II and III are true
- B.
Only II and III are true
- C.
Only III is true
- D.
None of them is true
Attempted by 106 students.
Show answer & explanation
Correct answer: C
Final answer: Only statement III is true.
Statement I (m < n): False. Underdetermined systems can be inconsistent. Example: with m = 1 and n = 2 the single equation 0·x1 + 0·x2 = 1 has no solution.
Statement II (m > n): False. Having more equations than variables does not guarantee inconsistency because equations may be dependent. Example: for m = 3 and n = 2 the system x + y = 2, 2x + 2y = 4, 3x + 3y = 6 is consistent.
Statement III (m = n): True. There exists at least one square system that has a solution: take the n×n identity matrix as the coefficient matrix and any right-hand side vector; the system is solvable.
Conclusion: only the third statement is guaranteed to be true in the sense that a square system can be constructed to have a solution; the first two statements are not universally true and counterexamples exist.