A linear system with n unknowns has coefficient-matrix rank r₁ and augmented-matrix rank r₂. The condition for the system to be consistent with a UNIQUE solution is:
Options:
(A) r₁ ≠ r₂ < n
(B) r₁ = r₂ = n
(C) r₁ = r₂ < n
(D) r₁ ≠ r₂ > n
Official Answer Key:
B
Step-by-Step Solution & Explanation
Consistency requires rank of coefficient matrix = rank of augmented matrix (r₁=r₂). A unique solution additionally requires this common rank to equal the number of unknowns n (no free variables); if r₁=r₂<n, infinitely many solutions result instead.
Practice This in Real CBT Exam Interface
Test yourself with the official TCS-iON palette, virtual calculator, and immediate performance analytics.