Describe the elements of M2(Z) that have multiplicative inverses.? Solution all the elements in M2(Z) are 2x2 matrices over Z Hence their determinant are also integers. det(AB) = det(A) det(B) if B is inverse of A then det(AB) = 1 if both A,B are M2(Z) and they are inverses of one other then det (A) = det(B) = +1 or -1 Say A = (a b) (c d) and ad-bc = +1 or -1 then B = (d -b) (-c a) Then AB = I Hence the only Matrices in M2(Z) are those whose determinant is +1 or -1.