Answer: rank(A) does not equal rank([A|B]).
- A rank(A) = rank([A|B]), the consistency condition
- B A is invertible, which guarantees a unique solution
- C rank(A) does not equal rank([A|B])
- D det(A) is not zero, the invertibility condition
Correct answer: C. rank(A) does not equal rank([A|B])
Explanation: Rouché-Capelli theorem: AX=B is consistent iff rank(A) = rank([A|B]). Adding B as a column raises the rank only if B is not in the column space of A, making the system inconsistent. Answer: rank(A) ≠ rank([A|B]).
Concept context
Evaluation of determinants, cofactors, adjoint, inverse of matrix, Cramer's rule and area applications. Always in board and JEE.