Three methods to find A^(-1):

1. Adjoint method: A^(-1) = adj$\frac{A}{det}$(A). Best for 2x2. Tedious for 3x3 but always works.

2. Row reduction: Augment [A | I] and row-reduce to [I | A^(-1)]. Best for 3x3 in exams.

3. Cayley-Hamilton: Find characteristic equation, then express A^(-1) in terms of A and I. Fastest when A^(-1) is asked alongside other powers of A.

Verification: Always check at least one entry of AA^(-1) to catch arithmetic errors. For 2x2, full verification takes 30 seconds and is worth it.

When inverse does NOT exist: det(A) = 0 (singular matrix). If a problem asks "for which values of k does A^(-1) exist?" -- set det(A) != 0 and solve.