Determinant of a 2×2 Matrix

Down the main diagonal, multiply; down the other, multiply and subtract. For the matrix with rows (3, 8) and (4, 6), D = 3·6 − 8·4 = 18 − 32 = −14. Small as it is, that number carries the matrix's whole character. Its absolute value is the area of the parallelogram spanned by the two rows, so it measures how much the matrix stretches the plane; its sign says whether the transformation preserves orientation or mirrors it; and a determinant of zero means the two rows are parallel, the parallelogram is flat, and the matrix has no inverse.

Determinants reached the West through Gabriel Cramer's 1750 Introduction à l'analyse des lignes courbes algébriques, where an appendix gave the rule that still bears his name: solve a linear system by taking ratios of determinants. For 2×2 systems Cramer's rule is genuinely convenient; for large ones it is a computational disaster, needing vastly more arithmetic than elimination. The determinant here can be solved backwards for any single entry — with D = 1, a = 2, b = 3 and c = 1, the missing entry must be d = (1 + 3)/2 = 2 — provided the entry diagonally opposite is non-zero, since it is that partner which multiplies the unknown.