Adjoint and inverse of a matrix

The essentials

The adjoint of a square matrix A, written adj A, is the transpose of the matrix of cofactors of A. Its key property is A(adj A) = (adj A)A = |A| I.

A square matrix A is invertible (non-singular) if there is a matrix B with AB = BA = I; then B = A⁻¹. Dividing the key property by |A| gives A⁻¹ = (1/|A|) adj A, which works only when |A| ≠ 0. A matrix with |A| = 0 is singular and has no inverse.

Key points

• For A = [[a, b], [c, d]], adj A = [[d, −b], [−c, a]]: swap the diagonal elements and change the signs of the others.
• (AB)⁻¹ = B⁻¹A⁻¹ and (Aᵀ)⁻¹ = (A⁻¹)ᵀ.
• |A⁻¹| = 1/|A|.
• For an n × n matrix, |adj A| = |A|ⁿ⁻¹.
• If a matrix satisfies an equation such as A² − 5A + 7I = O, multiplying by A⁻¹ gives the inverse directly.

Exam tips

• Always find |A| first: if it is 0, stop, because the inverse does not exist.
• Check your inverse quickly by multiplying A and A⁻¹ for one row and column.

Common mistakes

• Forgetting to transpose the cofactor matrix when forming adj A for a 3 × 3 matrix.
• Writing (AB)⁻¹ = A⁻¹B⁻¹; the order must be reversed.