Let $A=\left[\begin{array}{l}a\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}b\\ c\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}d\end{array}\right]$, be a 2x2 matrix where $a,b,c,d\in \{0,1\}$. The number of such matrices which have inverse is

$\det \left(A\right)=ad-bc$

Note that det(A) can take value –1, 0 or 1. We have

$\det \left(A\right)=1\iff ad=1,bc=0$

$\iff a=1,d=1$ or $(b=0,c=0,b=0;c=1$

$b=1,c=0)$

and $\det \left(A\right)=-1\iff ad=0$ or $bc=1$

This is also possible in 3 cases.

$\therefore A^{-1}$ exists in 6 cases.