Determinants

Step 1: Express the system of equations in matrix form.
We are given the system of linear equations: We can express this system of equations in matrix form as:
Step 2: Apply the condition for a unique solution.
For a system of linear equations to have a unique solution, the determinant of the coefficient matrix must be non-zero.
The determinant of the matrix is: We will compute this determinant:
Step 3: Solve for .
For a unique solution, the determinant must be non-zero.
Therefore, we set the determinant not equal to zero: Thus, the system has a unique solution if .
From the options, this corresponds to option (a), .
Step 4: Conclusion.
Therefore, the correct answer is , which corresponds to option (a).