Say I have a $3\times 3$ matrix called $\mathbf A$ and a column matrix vector $\mathbf v$ and another column matrix vector $\mathbf b$.
If I have the equation $\mathbf{Av}=\mathbf{b}$ where I know $\mathbf A$ and $\mathbf b$ can I solve for $\mathbf v$ by doing $\mathbf{A}^{-1}\mathbf{b}$?
If $Av=b$ and $A$ is invertible, then $v=A^{-1}b$. But even if $A^{-1}$ exists, calculating it might be a cumbersome task. In general, the problem of finding $v$ is the same as solving a linear system of equations. You don't need to calculate $A^{-1}$ to find $v$. There are several methods to do so, including for cases when $A$ is not invertible.
An interesting example: in MATLAB, instead of calculating $v$ by typing
v = inv(A)*b(it means "calculate $A^{-1}$ and then multiply it by $b$"), you should always typev = A\b(it means "solve $Av=b$, without necessarily calculating $A^{-1}$"). The later code is usually much faster.