Inverse of a 2×2 Matrix (Top-Left Element)

Inverting a 2×2 matrix is one of the few genuinely memorable recipes in linear algebra: swap the two diagonal entries, negate the off-diagonal pair, and divide everything by the determinant. The top-left entry of the inverse is therefore d/(ad − bc) — note that it depends on d, not on a. For M with rows (4, 7) and (2, 6) the determinant is 24 − 14 = 10, so the inverse's top-left entry is 6/10 = 0.6. Arthur Cayley set out the general theory in his 1858 A Memoir on the Theory of Matrices, the paper where matrix multiplication and matrix inverses first appear as an algebra in their own right rather than as bookkeeping for equations.

Two traps account for most errors. The first is forgetting to divide by the determinant at all, which leaves the adjugate matrix — right in pattern, wrong in scale. The second is a determinant of zero: such a matrix collapses the plane onto a line, information is destroyed, and no inverse exists at any price. The relation runs backwards too, since it is linear in each entry taken alone: given an inverse entry of −2.5 with b = 3, c = 4 and d = 5, the missing entry must be a = (5 − 30)/(−12.5) = 2.