The adjoint (adjugate) of a matrix A, denoted adj(A), is the transpose of the cofactor matrix. For a 2x2 matrix [[a,b],[c,d]], the adjoint is [[d,-b],[-c,a]] (swap diagonal, negate off-diagonal).

Computing adj(A) for 3x3:

1. Find all 9 cofactors $C_{ij}$ = (-1)^(i+j) * $M_{ij}$
2. Arrange in a matrix
3. TRANSPOSE the cofactor matrix

Key identities (all for n x n):

• A * adj(A) = adj(A) * A = det(A) * I
• det(adj(A)) = det(A)^(n-1)
• adj(adj(A)) = det(A)^(n-2) * A
• adj(AB) = adj(B) * adj(A) [reversal!]
• adj(kA) = k^(n-1) * adj(A)

Finding A^(-1): Method 1: adj$\frac{A}{det}$(A) -- best for 2x2 Method 2: Row-reduce [A|I] to [I|A^(-1)] -- best for 3x3 Method 3: Cayley-Hamilton -- best when $A^n$ is also needed

When inverse doesn't exist: det(A) = 0 makes A singular. For parametric matrices "find k such that A^(-1) exists": set det(A) != 0 and solve.

Properties of inverse: (AB)^(-1) = B^(-1)A^(-1), ($A^T$)^(-1) = (A^(-1))^T, ($A^n$)^(-1) = (A^(-1))^n