The correct answer is $\boxed{\text{A}}$.
The trace of a matrix is the sum of its eigenvalues, and the determinant of a matrix is the product of its eigenvalues. If the trace of a matrix is positive and the determinant of the matrix is negative, then at least one of its eigenvalues must be negative. This is because the trace and determinant of a matrix are both invariants of the matrix, meaning that they do not change when the matrix is transformed by a similarity transformation. Therefore, if the trace and determinant of a matrix have different signs, then the matrix must have at least one eigenvalue with a negative real part.
The other options are incorrect. Option $\text{B}$ is incorrect because it is possible for a matrix to have a positive trace and negative eigenvalues. For example, the matrix $$A = \begin{bmatrix} 1 & 2 \\ 0 & -1 \end{bmatrix}$$ has a trace of 1 and a determinant of -1, but its eigenvalues are both negative. Option $\text{C}$ is incorrect because it is possible for a matrix to have a positive determinant and negative eigenvalues. For example, the matrix $$B = \begin{bmatrix} 1 & 0 \\ 0 & -2 \end{bmatrix}$$ has a determinant of 2 and a negative eigenvalue. Option $\text{D}$ is incorrect because it is possible for a matrix to have a positive product of the trace and determinant and negative eigenvalues. For example, the matrix $$C = \begin{bmatrix} 1 & 2 \\ 0 & -3 \end{bmatrix}$$ has a product of the trace and determinant of 3 and a negative eigenvalue.