A matrix is invertible if and only if its determinant is non-zero. If A had determinant 0, then . but which has determinant 1, not 0. Therefore det(A) is not 0. Same thing for B. Then det(AB)= det(A)det(B) is non-zero.
I chose those four matrices because they are the "standard basis" for the vector space of 2 by 2 matrices. Every such matrix can be written as a linear combination of them:
so they "represent" all 2 by 2 matrices.