Taking a simple example we can get a clear idea.
Let matrix A be
Here trace(A)=4 and determinant(A)=80.
As matrix is diagonal matrix the principal diagonal elements are the eigenvalues for matrix A.
Even though a trace of a matrix is positive we have two negative eigenvalues(i.e. -2,-4). So option B is false.
Even though determinant of a matrix is positive we have two negative eigenvalues(i.e. -2,-4). So option C is false.
the product of trace and determinant is 320 which is positive still we have two negative eigenvalues(i.e. -2,-4). So option D is false.
Now if the determinant is negative then at least one eigenvalue of the matrix is negative because the product of eigenvalues of a given matrix is equal to the determinant of a given matrix.
So OPTION A is True.