Every $n \times n$ matrix is ​the limit of a sequence of invertible matrices $n \times n$.

Question

Every $n \times n$ matrix is ​​the limit of a sequence of invertible matrices $n \times n$.

Someone has a clue how to solve this question, honestly, I have no idea how to work with sequence of arrays, because I have never seen something like this before.

Answer 1

Let $p_A(x) = \det(A - xI)$ be the characteristic polynomial of $A$. This is a polynomial of degree $n$ so it has at most $n$ roots. Hence you can choose a sequence $x_n \to 0$ such that $p_A(x_n) \neq 0$. Then $A - x_nI$ is a sequence of matrices which converges to $A$ and since $p_A(x_n) = \det(A - x_nI) \neq 0$, each of the matrices $A - x_n I$ is invertible.