Determinant and inverse
det A scalar from a square matrix; A is invertible iff det A ≠ 0.
2×2: det A = ad − bc. A−1 = (1/det A)(d −b; −c a). 3×3: cofactor expansion. Used to solve linear systems Ax = b ⇒ x = A−1 b.
Worked examples
A = (3 1; 2 1): det = 1; A−1 = (1 −1; −2 3).
3×3 cofactor expansion: choose any row/column.
Singular matrix: det = 0; no inverse; system has no unique solution.
Frequently asked questions
Why does det = 0 imply singular?
Geometrically: the transformation collapses space. Algebraically: rows linearly dependent.
Cramer's rule?
Solving Ax = b using determinants. Useful for 2×2; less so for larger.