If $\|X(t)\|\leq M$, does this imply that $det(X(t))$ is bounded?

Question

I am wondering if the following is true:

If you are given a matrix $X(t)$ (that depends on the positive real variable $t$) which is bounded (i.e, $\|X(t)\|\leq M$ for all $t$. Can you conclude that $\det(X(t))$ is also bounded? Can you also conclude that $\lim_{t\to \infty}\det(X(t))$ is finite? Thanks!

Answer 1

Yes, the determinant is a continuous function of the entries of a matrix

Answer 2

Warning: $\det(X(t))$ is bounded (answer by Student), but this does not guarantee the existence of $\lim_{t\to\infty}\det(X(t))$. Trivial $1\times 1$ example: $(\sin(t))$.