Question

The system AX=B has a unique solution if

• A. \( \rho(A) = \rho(A|B)<\text{No. of Variables} \)
• B. \( \rho(A)<\rho(A|B) = \text{No. of Variables} \)
• C. \( \rho(A)<\rho(A|B)<\text{No. of Variables} \)
• D. \( \rho(A) = \rho(A|B) = \text{No. of Variables} \)

Hint:

System of Linear Equations AX=B (Rouché-Capelli). Let rank(A)=\(r_A\), rank(A|B)=\(r_{AB\), No. Variables=N. - No Solution if \(r_A \neq r_{AB\). - Unique Solution if \(r_A = r_{AB = N\). - Infinite Solutions if \(r_A = r_{AB<N\).

Answer 1

The Correct Option is D

The consistency and nature of solutions for a system of linear equations represented by AX=B can be determined using the Rouché-Capelli theorem, which relates the rank of the coefficient matrix (A) and the rank of the augmented matrix (A|B)
Let N be the number of variables (columns in A)
- No Solution (Inconsistent): If rank(A) \(\neq\) rank(A|B)
(\(\rho(A)<\rho(A|B)\))
- Solution Exists (Consistent): If rank(A) = rank(A|B) = r
- Unique Solution: If r = N (rank equals the number of variables)
- Infinite Solutions: If r<N (rank is less than the number of variables)
The question asks for the condition for a unique solution
This requires the system to be consistent (\(\rho(A) = \rho(A|B)\)) and the rank to be equal to the number of variables (\(r = N\))
Therefore, the condition is \(\rho(A) = \rho(A|B) = \text{No
of Variables}\)