Let A=a    bc    d, be a 2x2 matrix where a,b,c,d∈{0,1}. The number of such matrices which have inverse is

det⁡(A)=ad−bcNote that det(A) can take value –1, 0 or 1. We havedet⁡(A)=1⇔ad=1,bc=0⇔a=1,d=1 or  (b=0,c=0,b=0;c=1b=1,c=0)and det⁡(A)=−1⇔ad=0 or bc=1This is also possible in 3 cases.∴ A−1 exists in 6 cases.