For a system of linear equations $Ax = b$, where $A$ is an $m \times n$ matrix with rank $r$, which of the following statements is/are correct?
Options:
(A) If $\operatorname{rank}(A) = \operatorname{rank}([A|b]) = n$, the system has a unique solution
(B) If $\operatorname{rank}(A) < \operatorname{rank}([A|b])$, the system is inconsistent and has no solution
(C) If $\operatorname{rank}(A) = \operatorname{rank}([A|b]) < n$, the system has infinitely many solutions
(D) A homogeneous system $Ax = 0$ always has at least one solution (the trivial solution)
Answer Key:
A, B, C, D
Step-by-Step Solution & Explanation
By the Rouché–Capelli theorem:
1. A system $Ax = b$ is consistent if and only if $\operatorname{rank}(A) = \operatorname{rank}([A|b])$.
2. If consistent and rank $r = n$ (number of unknowns), there is a unique solution (A is true).
3. If $\operatorname{rank}(A) < \operatorname{rank}([A|b])$, no solution exists (B is true).
4. If consistent and $r < n$, there are $(n-r)$ linearly independent free variables, producing infinitely many solutions (C is true).
5. For a homogeneous system $Ax=0$, $x=0$ is always a solution (trivial), so it is always consistent (D is true).