Matrices

Step 1: Constructing the Coefficient Matrix The given system of equations corresponds to the coefficient matrix: For the system to have a unique solution, the determinant of the matrix must be non-zero, i.e., .
Step 2: Computing the Determinant Expanding along the first row: Calculating the minors: Substituting these:
Step 3: Condition for Unique Solution For a unique solution:
Step 4: Determining Since the determinant only depends on , can take any real value: Thus, the correct answer is:

Step 1: Using the Determinant Property For any matrix , the determinant of its adjugate matrix is given by: Since and are matrices, applying this formula:
Step 2: Applying the Property to Product Matrices Since , and using determinant properties: Thus, the correct answer is:

We are given that the matrices and satisfy the condition , and we are tasked with finding the matrix . To solve this, we must analyze the structure of the matrices and use the property of commutative matrices. First, let's test the given options by multiplying with each of the possible matrices and check if the result satisfies the condition . After testing each option, we find that the matrix that satisfies is: Thus, the correct answer is option (4).