Can we interchange the limit with determinant?

Question

Let $\mathbf{A}(\rho)$ be a matrix which is a function of $\rho$. Under what condition does the following equation hold? $$\lim_{\rho \rightarrow \infty} \det [ \mathbf{A}(\rho) ] = \det \left[ \lim_{\rho \rightarrow \infty} \mathbf{A}(\rho) \right], \tag{1}$$ where $\det [ \mathbf{A}(\rho) ]$ is the determinant of the matrix $\mathbf{A}(\rho)$.

Answer 1

Assuming $\lim\limits_{\rho \to \infty} \mathbf{A}(\rho)$ exists, that is always true, because $\det: M_{n \times n}(\Bbb{R}) \to \Bbb{R}$ is a continuous function.