Consider $\mathbf{B}$ be a matrix invertable modulo a prime number $p$. Is it always possible to say that $\mathbf{B}$ is always invertable modulo $p^\alpha,\, \alpha \in \mathbb{N}$.
It seems that by Hensel lifting, If $\mathbf{B}=\mathbf{I}-\mathbf{A}$ this fact is true when the characteristic polynomial of the matrix $\mathbf{A}$ be primitive modulo $p$, but in general I don't know how to apply this.
A matrix over a commutative ring is invertible if and only if its determinant is invertible.
Thus if an integral matrix is invertible modulo $p$ then its determinant is coprime to $p$ and thus to $p^a$ for any positive $a$ and thus it is invertible modulo $p^a$.