For a system of $m$ linear equations in $n$ variables represented as $A X = B$, where $A$ is the coefficient matrix and $[A|B]$ is the augmented matrix, which of the following statements is/are CORRECT?
Options:
(A) If $\operatorname{rank}(A) \neq \operatorname{rank}([A|B])$, the system has no solution (inconsistent)
(B) If $\operatorname{rank}(A) = \operatorname{rank}([A|B]) = n$, the system has a unique solution
(C) If $\operatorname{rank}(A) = \operatorname{rank}([A|B]) = r < n$, the system has infinitely many solutions with $(n - r)$ linearly independent solutions
(D) For a homogeneous system $A X = 0$, a non-trivial solution exists if and only if $\operatorname{rank}(A) < n$
Answer Key:
A, B, C, D
Step-by-Step Solution & Explanation
According to the Rouché–Capelli theorem for linear systems $A X = B$:
1. A system is consistent if and only if $\operatorname{rank}(A) = \operatorname{rank}([A|B])$. If ranks differ, it is inconsistent (Option A is true).
2. If $\operatorname{rank}(A) = \operatorname{rank}([A|B]) = n$ (number of variables), exactly one unique solution exists (Option B is true).
3. If $\operatorname{rank}(A) = \operatorname{rank}([A|B]) = r < n$, there are infinitely many solutions with $n - r$ free parameters (Option C is true).
4. For homogeneous systems $A X = 0$, non-trivial (non-zero) solutions exist if and only if $\operatorname{rank}(A) < n$, corresponding to $\det(A) = 0$ for square systems (Option D is true).