Adjoint and inverse

The adjoint of A, written adj A, is the transpose of the matrix of cofactors: find every cofactor, put each in its element's place, then turn rows into columns.

2 × 2 shortcut: for [a, b; c, d], adj A = [d, –b; –c, a]. Swap the two diagonal entries and change the sign of the other two.

The key fact: A(adj A) = (adj A)A = |A| I. Dividing by |A| gives the inverse:

A⁻¹ = (1/|A|) adj A

This only works when |A| ≠ 0. So:

• |A| = 0: A is singular and has no inverse.
• |A| ≠ 0: A is non-singular and has exactly one inverse.

Example: A = [3, 1; 2, 1]. |A| = 3 – 2 = 1. adj A = [1, –1; –2, 3], so A⁻¹ = [1, –1; –2, 3]. Check: A A⁻¹ = [3 – 2, –3 + 3; 2 – 2, –2 + 3] = I.

For a 3 × 3 matrix: first find |A| (stop if it is 0), then all nine cofactors, then transpose, then divide by |A|. Keep the signs careful: most mistakes happen there.

Useful results:

• |AB| = |A||B| and (AB)⁻¹ = B⁻¹A⁻¹.
• |A⁻¹| = 1/|A|.
• |adj A| = |A|ⁿ⁻¹ for an n × n matrix, so |adj A| = |A|² for 3 × 3.

If A satisfies an equation such as A² – 5A + 7I = O, multiply through by A⁻¹: A – 5I + 7A⁻¹ = O, so A⁻¹ = (5I – A)/7. No cofactors needed.