Question

If the matrix \(A\) is of order \(3 \times 3\) and the system of equations \(AX=B\) has a unique solution, what can be concluded about the determinant of \(A\)?

• A. The determinant of \(A\) is zero
• B. The determinant of \(A\) is non-zero
• C. The determinant of \(A\) must be \(1\) only
• D. The determinant of \(A\) cannot be negative

Answer 1

The Correct Option is B

Concept: 
For a system of linear equations: \[ AX=B \] a unique solution exists only when the coefficient matrix \(A\) is invertible. 

Step 1: The given system is: \[ AX=B \] 

Step 2: It is given that the system has a unique solution. 

Step 3: A system has a unique solution if the coefficient matrix \(A\) is non-singular. 

Step 4: A matrix is non-singular if: \[ |A|\neq 0 \] 

Step 5: Therefore, the determinant of \(A\) must be non-zero. \[ \boxed{\text{The determinant of \(A\) is non-zero}} \]