For a non-homogeneous system of linear equations $Ax = b$ where $A$ is an $m \times n$ matrix and $b \neq 0$, which of the following statements is/are TRUE?
Options:
(A) The system is consistent if and only if $\operatorname{rank}(A) = \operatorname{rank}([A|b])$
(B) If $\operatorname{rank}(A) = \operatorname{rank}([A|b]) = n$, the system has a unique solution
(C) If $\operatorname{rank}(A) = \operatorname{rank}([A|b]) < n$, the system has infinitely many solutions
(D) If $\operatorname{rank}(A) < \operatorname{rank}([A|b])$, the system has no solution
Answer Key:
A, B, C, D
Step-by-Step Solution & Explanation
By Rouché-Capelli Theorem:
1. A system $Ax = b$ has at least one solution (is consistent) iff $\operatorname{rank}(A) = \operatorname{rank}([A|b])$. (A is true)
2. If the rank equals the number of variables $n$, the solution is unique. (B is true)
3. If the rank is strictly less than $n$, there exist $n - \operatorname{rank}(A)$ free parameters, giving infinitely many solutions. (C is true)
4. If $\operatorname{rank}(A) < \operatorname{rank}([A|b])$, the equations are contradictory and no solution exists. (D is true)
All choices A, B, C, D are correct.