Question

A non-homogeneous system of linear equation \(AX=B\) of \(n\) unknowns is called consistent if:

• A. \(\operatorname{Rank}(A)=n\)
• B. \(\operatorname{Rank}(A:B)=n\)
• C. \(\operatorname{Rank}(A:B)=\operatorname{Rank}(A)=0\)
• D. \(\operatorname{Rank}(A)<\operatorname{Rank}(A:B)\)

Hint:

For consistency, remember the general condition \(\operatorname{Rank}(A)=\operatorname{Rank}(A:B)\).

Answer 1

The Correct Option is B

Concept:
A system of linear equations is consistent if it has at least one solution.

Step 1: General condition.
For a non-homogeneous system: \[ AX=B \] the system is consistent when: \[ \operatorname{Rank}(A)=\operatorname{Rank}(A:B) \]

Step 2: Unique solution case.
If the number of unknowns is \(n\) and: \[ \operatorname{Rank}(A)=\operatorname{Rank}(A:B)=n \] then the system has a unique solution and is definitely consistent.

Step 3: Select the suitable option.
Among the given options, option (B) represents the full-rank augmented condition used here. \[ \therefore \text{Correct Answer is (B)} \]