How do I find the inverse of a 2x2 matrix and when does it not exist?

For a 2x2 matrix with rows (a, b) and (c, d), first find the determinant: det = ad − bc. The inverse exists only if det is not zero. If det = 0 the matrix is called singular and has no inverse. When det is non-zero, swap a and d, change the signs of b and c, then divide every entry by det. So the inverse is (1/det) times the matrix with rows (d, −b) and (−c, a). For example if the matrix has rows (2, 1) and (5, 3), det = 6 − 5 = 1, and the inverse has rows (3, −1) and (−5, 2). Always verify by checking that the matrix times its inverse gives the identity matrix.