Prove that if $n\times n$ matrix $A$ is nonsingular, then it is invertible. I know that nonsingularity means having unique soln. $x$ for every RHS b in the system $Ax= b$. But how do I use this definition to come up with a proof for the problem ? I really have tried but to no avail.

Recall that
$$\mathbf{M}^{-1}=\frac{1}{\det(\mathbf{M})}\operatorname{adj}(\mathbf{M})$$
Simply show that $\operatorname{adj}(\mathbf{M})$ always exists (this is not difficult), and thus $\mathbf{M}^{-1}$ always exists if $\det(\mathbf{M}) \neq 0$.
