If matrix $A_k$ is invertible for all $k$ but $\lim_{k \rightarrow \infty} A_k$ is not invertible, is $\lim_{k \rightarrow \infty} A_k = 0$?

Question

Let $\{A_k\}_{k=1}^{\infty}$ be a sequence of real $n \times n$ matrices. Suppose $A_k$ is invertible for all $k \in \mathbb{N}$, and that $\lim_{k \rightarrow \infty} A_k$ exists but is NOT invertible. Is $\lim_{k \rightarrow \infty} A_k = 0$?

I have a hunch this follows from continuity of the matrix inverse, but can't quite work out a proof. Or maybe it's not true at all!

Answer 1

No. First row of a $2 \times 2~A_k$ is $\left (\frac 1k, 1 \right )$, second row is $\left (0, \frac 1k \right )$.