Let A be an m x n matrix, where m < n. Consider the system of linear equations Ax = b. Which of the following statements are TRUE about the system of linear equations Ax = b? 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. 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.

GATE 2016 · Engineering Mathematics · System of Equations · medium

Answer: Only Statement III is true. Answer: C.

1. Evaluate Statement I: Statement I claims: m < n => all systems Ax=b are consistent. Counterexample: take m=1, n=2, A=[1,1], b=[1] gives x+y=1 (consistent). But take A=[1,1;1,1] (m=2, n=2 < ... wait). For m=1, n=2: A=[1,1], b=0: x+y=0 consistent. Actually any 1-equation system is consistent (just pick values). Better: m=2, n=3: A=[[1,1,1],[1,1,1]], b=[1;2]: rank(A)=1, rank([A|b])=2 => inconsistent! So Statement I is FALSE.
2. Evaluate Statement II: Statement II claims: m >= n => no system has a solution. Counterexample: m=n=2, A=I (identity), b=[1;1]: x=1, y=1 is a solution. Statement II is FALSE.
3. Evaluate Statement III: Statement III claims: m = n => there exists a system Ax=b with a solution. Example: take A=I_n, b=0. Then x=0 is a solution. So at least one such system exists. Statement III is TRUE.
4. Identify the correct answer: I is false, II is false, III is true. Answer: C (Only III is true).