Allied & ICAR Exam Bank • ALLIED_2631MSQ • 2 Marks
Section 1: Engineering Mathematics
Linear Algebra
Subtopic: Solutions of linear equations
Question Statement:
For an inconsistent system of linear equations $A x = b$, which of the following statements are TRUE?
Options:
(A) $\text{rank}(A) < \text{rank}([A|b])$
(B) There is no vector $x$ satisfying $A x = b$
(C) The vector $b$ does not lie in the column space of $A$
(D) $\det(A) \neq 0$ for square $A$
Answer Key:
A, B, C
Step-by-Step Solution & Explanation
If $A x = b$ is inconsistent, $b$ is outside $\text{col}(A)$, so augmented rank exceeds matrix rank, and no solution exists. If $\det(A) \neq 0$, the system is always consistent with unique solution $x = A^{-1} b$, so (D) is false.