Determinants

Step 1: Determinant properties.
When a matrix is equal to a scalar multiplied by matrix , i.e., , the determinant of is related to the determinant of by the property: where is the order of the square matrix. For a 3x3 matrix, .
Step 2: Conclusion.
Thus, . Therefore, the correct answer is (4) .

Step 1: Determining the condition for a non-trivial solution.
For a homogeneous system of linear equations to have a non-trivial solution, the determinant of the coefficient matrix must be zero. The coefficient matrix is: The determinant of this matrix is: For a non-trivial solution, the determinant must be zero:
Step 2: Conclusion.
Therefore, the value of for which the system has a non-trivial solution is (1) .