Let A = [a<sub>ij</sub>] and B = [b<sub>ij</sub>] be two square matrices of order n and det(A) denote the determinant of A. Then, which of the following is not correct:
Step-by-step Solution:
Option A: If \( A \) is a diagonal matrix, then \( \text{det}(A) = a_{11} a_{22} \ldots a_{nn} \). - This statement is **correct**. The determinant of a diagonal matrix is the product of its diagonal entries. So, if \( A \) is a diagonal matrix, \( \text{det}(A) \) is indeed the product of the diagonal elements \( a_{11}, a_{22}, \ldots, a_{nn} \). \[\] Option B: \( \text{det}(AB) = \text{det}(A) \text{det}(B) \). - This statement is correct. The determinant of the product of two matrices is the product of their determinants. This property holds for square matrices of the same order. \[\] Option C: \( \text{det}(cA) = c [\text{det}(A)] \). - This statement is incorrect. The correct formula is: \[ \text{det}(cA) = c^n \cdot \text{det}(A) \] where \( c \) is a scalar and \( n \) is the order of the matrix. So, the determinant of \( cA \) is \( c^n \) times the determinant of \( A \), not just \( c \) times the determinant. \[\] Option D: \( \text{det}(A) = \text{det}(A^T) \), where \( A^T \) denotes the transpose of matrix \( A \). - This statement is correct. The determinant of a matrix is equal to the determinant of its transpose. \[\] Conclusion: The incorrect statement is Option C: \( \text{det}(cA) = c [\text{det}(A)] \). The correct formula is \( \text{det}(cA) = c^n \cdot \text{det}(A) \).