Let Ax = b be a system of linear equations where A is an m x n matrix and b is a m x 1 column vector and x is an n x 1 column vector of unknowns. Which of the following is false?
A. The system has a solution only if both A and the augmented matrix [A|b] have the same rank.
B. If m < n and b is the zero vector, then the system has infinitely many solutions.
C. If m = n and b is a non-zero vector, then the system has a unique solution.
D. The system will have only a trivial solution when m = n, b is the zero vector, and rank(A) = n.
GATE 1996 · Engineering Mathematics · System of Equations · medium
Answer: Statement C is false: m=n and b!=0 does not guarantee a unique solution.
Verify Statement A: This is the Rouche-Capelli theorem. Statement A is TRUE.
Verify Statement B: With m < n and b=0 (homogeneous): rank(A) <= m < n, so nullity >= 1, meaning infinitely many solutions. Statement B is TRUE.
Evaluate Statement C (candidate for false): m=n does not imply rank(A)=n. If det(A)=0, the system may have no solution (b not in col(A)) or infinitely many solutions (b in col(A)). Example: A=[[1,2],[2,4]], b=[1,0]: no solution. Statement C is FALSE.
Verify Statement D: With b=0: x = A^{-1}*0 = 0. Only the trivial solution exists. Statement D is TRUE.