Essential Adjoint Formulas:
| Formula | Condition |
|---|---|
| adj(A) = (cofactor matrix)^T | Always |
| A * adj(A) = det(A) * I | Always |
| adj(A) * A = det(A) * I | Always |
| det(adj(A)) = det(A)^(n-1) | n x n matrix |
| adj(adj(A)) = det(A)^(n-2) * A | det(A) != 0 |
| adj(AB) = adj(B) * adj(A) | Note the reversal |
| adj(kA) = k^(n-1) * adj(A) | n x n matrix |
| adj() = (adj(A))^T | Always |
| adj(A^(-1)) = (adj(A))^(-1) = (A) | det(A) != 0 |
For 2x2 matrix [[a,b],[c,d]]: adj = [[d,-b],[-c,a]] (swap diagonal, negate off-diagonal)
For 3x3: Must compute all 9 cofactors and transpose. No shortcut.