Linear Algebra

For a $2 \times 2$ matrix $M$, $\det(c M) = c^2 \det(M)$. Thus $\det(2 A^T A^{-1}) = 2^2 \det(A^T) \det(A^{-1}) = 4 \times \det(A) \times \frac{1}{\det(A)} = 4$.