Inverse of a matrix

When an inverse exists

• A square matrix A has an inverse A⁻¹ only if det A ≠ 0; such a matrix is called non-singular.
• If det A = 0 the matrix is singular and has no inverse.
• Definition: AA⁻¹ = A⁻¹A = I.

The 2×2 formula

• For A = [[a, b], [c, d]] with det A ≠ 0: A⁻¹ = (1/det A) × [[d, −b], [−c, a]].
• The matrix [[d, −b], [−c, a]] is the adjugate (transpose of the cofactor matrix).
• Example: A = [[3, 5], [2, 4]], det A = 2 → A⁻¹ = ½ [[4, −5], [−2, 3]] = [[2, −2.5], [−1, 1.5]].
• Check: A A⁻¹ = [[3·2 + 5·(−1), 3·(−2.5) + 5·1.5], [2·2 + 4·(−1), 2·(−2.5) + 4·1.5]] = [[1, 0], [0, 1]] = I.

Useful identities

• (A⁻¹)⁻¹ = A  ·  (AB)⁻¹ = B⁻¹A⁻¹  ·  (Aᵀ)⁻¹ = (A⁻¹)ᵀ  ·  det(A⁻¹) = 1/det A.