Show that the matrix $\displaystyle A=\left[\begin{matrix}a&b\\c&d\end{matrix}\right]$ is nonsingular if and only if ad-bc≠0. If this condition holds, show that $\displaystyle \left[\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right]$

I think what I want to do is show that $\displaystyle \left[\begin{matrix}a&b\\c&d\end{matrix}\right]$ is row equivalent to $\displaystyle I_2$. When I was doing this I got stuck at $\displaystyle \left[\begin{matrix}ca&da\\0&1\end{matrix}\right]$