Q:

Showing a $2\times2$ matrix is invertible

Let ${A}$ be a $2 \times 2$ matrix. For every two-dimensional vector ${v}$, there exists a two-dimensional vector ${w}$ such that
${A} {w} = {v}.$
Show that ${A}$ is invertible.
I have no idea on how to start this, but I have a hint:
Which two vectors ${v}$ are we most interested in if we are trying to find the matrix ${B}$ such that ${A}{B}= {I}$?
Any help is appreciated!

A:

HINT: Let $v_1$, $v_2$ be  column vectors of $I$ (identity matrix).

