Suppose that A and B are n×n matrices. Check the true statements below: A. If det(A) is zero, then two rows or two columns are the same, or a row or a column is zero.B. det(AT)=(−1)det(A)C. If two row interchanges are made in sucession, then the determinant of the new matrix is equal to the determinant of the original matrix.D. The determinant of A is the product of the diagonal entries in A

A. This statement is not always true. While it is true that if a matrix has two identical rows or columns, or a row or column of zeros, its determinant will be zero, the converse is not necessarily true. A matrix can have a determinant of zero without having identical rows or columns or a row or column of zeros. For example, consider the matrix [[1, 2], [2, 4]]. Its determinant is zero, but it does not have identical rows or columns or a row or column of zeros.

B. This statement is false. The determinant of the transpose of a matrix is equal to the determinant of the original matrix, not its negative. So, det(AT) = det(A).

C. This statement is true. The determinant changes sign when two rows are interchanged. So, if two row interchanges are made in succession, the determinant of the new matrix is indeed equal to the determinant of the original matrix.

D. This statement is not always true. The determinant of a matrix is not necessarily the product of its diagonal entries. This is only true for diagonal and triangular matrices. For other matrices, the determinant is calculated using a more complex formula.