Matrices

Step 1: Analyze statement (A). We know the property relating the determinant of the adjoint of a matrix A (of order n) to the determinant of A: . For a matrix, , so . If , then . Therefore, statement (A) is TRUE. Step 2: Analyze statement (B). Let be a diagonal matrix of order 3. The cofactor matrix C is . The adjoint of A is the transpose of the cofactor matrix: . This is also a diagonal matrix. Therefore, statement (B) is TRUE. Step 3: Analyze statement (C). Let A and B be two upper triangular matrices. This means for and for . Consider the element where . . The term is non-zero only if . The term is non-zero only if . For a term to be non-zero, we need and , which implies . However, we are considering the case where . It is impossible to satisfy when . Therefore, every term in the sum must be zero, meaning for . Thus, the product AB is also an upper triangular matrix. Therefore, statement (C) is TRUE. Step 4: Analyze statement (D). Recall the property for the adjoint of a product of matrices: The statement given is . Since matrix multiplication (and thus multiplication of adjoints) is generally not commutative, the given statement is incorrect. Therefore, statement (D) is FALSE. Step 5: Identify the false statement. Based on the analysis, statement (D) is the false statement.

Step 1: Recall the property of determinants involving scalar multiplication. For any matrix A and any scalar , the determinant of the matrix is given by: or using the notation from the question: Step 2: Identify the values of and the scalar in this problem. The matrix A is a matrix, so . The scalar multiple is 3, so . Step 3: Apply the property to the specific case . Using the formula with and : Step 4: Compare this result with the given equation . We have: Step 5: Solve for k. Since we are given that , we can divide both sides of the equation by : So, the value of k is 27. Step 6: Compare the result with the given options. The calculated value matches option (D).

Step 1: Recall the definition of a symmetric matrix.
A matrix is symmetric if
Step 2: Take transpose of .

Step 3: Conclusion.
The transpose of is equal to .