Step 1: Understanding the Concept:
This question relates the existence of a unique solution for a system of linear equations to the properties of its coefficient matrix. The system of equations is given by $AX = B$, where A is the coefficient matrix, X is the vector of variables, and B is the constant vector.
Step 2: Detailed Explanation:
According to the Cramer's rule and the properties of linear systems:
A system of linear equations $AX=B$ has a unique solution if and only if the coefficient matrix A is invertible (non-singular).
A square matrix A is invertible if and only if its determinant is non-zero ($\det(A) \neq 0$).
Given that the system has a unique solution, it directly implies that the matrix A must be invertible. Therefore, the determinant of A must be non-zero.
Step 3: Analyzing the Options:
(A) If the determinant of A is zero, the matrix is singular. In this case, the system $AX=B$ has either no solution or infinitely many solutions, but not a unique solution. So, this is incorrect.
(B) If the determinant of A is non-zero, the matrix is invertible, which is the necessary and sufficient condition for a unique solution. This is correct.
(C) The determinant can be any non-zero value, not just 1. For example, if $\det(A) = 2$, the system still has a unique solution. So, this is incorrect.
(D) The determinant can be any non-zero real number, including negative numbers. A negative determinant does not prevent the existence of a unique solution. So, this is incorrect.
Step 4: Final Answer:
For the system of equations $AX=B$ to have a unique solution, the determinant of the matrix A must be non-zero. Hence, option (B) is the correct conclusion.