Question

The system of equations \( AX = B \), has a unique solution if ____ .

• A. Rank of \( A \) = Rank of \( [A:B] \) = \( n \)
• B. Rank of \( A \) = Rank of \( [A:B] \) \(<\) \( n \)
• C. Rank of \( A \neq \) Rank of \( [A:B] \)
• D. Rank of \( A<\) Rank of \( [A:B] \)

Hint:

Remember that for a unique solution, you need two things: the system must be consistent (ranks of \( A \) and \( [A:B] \) are equal), and there must be exactly one free variable (rank of \( A \) equals the number of unknowns).

Answer 1

The Correct Option is A

Step 1: Understand the system of linear equations \( AX = B \).
Here, \( A \) is the coefficient matrix, \( X \) is the column vector of unknowns, and \( B \) is the column vector of constants. The augmented matrix of the system is denoted by \( [A:B] \), which is formed by appending the column vector \( B \) to the coefficient matrix \( A \). Let the number of unknowns (the size of the vector \( X \)) be \( n \). Step 2: Recall the Rank-Nullity Theorem and its implications for the existence and uniqueness of solutions.
The existence and uniqueness of solutions to the system \( AX = B \) are determined by the ranks of the coefficient matrix \( A \) and the augmented matrix \( [A:B] \). Existence of a solution: A system of linear equations \( AX = B \) has at least one solution if and only if the rank of the coefficient matrix \( A \) is equal to the rank of the augmented matrix \( [A:B] \). That is, \( \text{Rank}(A) = \text{Rank}([A:B]) \). If \( \text{Rank}(A)<\text{Rank}([A:B]) \), the system is inconsistent and has no solution. Uniqueness of a solution: If a solution exists (i.e., \( \text{Rank}(A) = \text{Rank}([A:B]) \)), then the solution is unique if and only if the rank of the coefficient matrix \( A \) is equal to the number of unknowns \( n \). That is, \( \text{Rank}(A) = n \). If \( \text{Rank}(A) = \text{Rank}([A:B])<n \), the system has infinitely many solutions. Step 3: Combine the conditions for existence and uniqueness to find the condition for a unique solution.
For the system \( AX = B \) to have a unique solution, two conditions must be met:
1. A solution must exist, which requires \( \text{Rank}(A) = \text{Rank}([A:B]) \).
2. The solution must be unique, which requires \( \text{Rank}(A) = n \), where \( n \) is the number of unknowns.
Combining these two conditions, the system \( AX = B \) has a unique solution if and only if \( \text{Rank}(A) = \text{Rank}([A:B]) = n \). Step 4: Evaluate the given options.
Option 1: Rank of \( A \) = Rank of \( [A:B] \) = \( n \). This satisfies both conditions for existence and uniqueness of a solution.
Option 2: Rank of \( A \) = Rank of \( [A:B] \)<\( n \). A solution exists, but it is not unique (infinitely many solutions).
Option 3: Rank of \( A \neq \) Rank of \( [A:B] \). No solution exists (inconsistent system).
Option 4: Rank of \( A<\) Rank of \( [A:B] \). No solution exists (inconsistent system).
Step 5: Select the correct answer.
The system of equations \( AX = B \) has a unique solution if Rank of \( A \) = Rank of \( [A:B] \) = \( n \).