| Type | Definition | Determinant | Eigenvalues | Key Property |
|---|---|---|---|---|
| Symmetric | A = A^T | Any | All real | Always diagonalizable |
| Skew-Symmetric | A = -A^T | 0 (odd n) | 0 or purely imaginary | Diagonal entries = 0 |
| Orthogonal | AA^T = I | +1 or -1 | lambda | |
| Idempotent | A^2 = A | 0 or 1 | 0 or 1 | tr(A) = rank(A) |
| Involutory | A^2 = I | +1 or -1 | +1 or -1 | A^(-1) = A |
| Nilpotent | A^k = O | 0 | All zero | tr(A) = 0 |
Key distinctions for JEE:
- Orthogonal vs Involutory: Both have A appearing in their inverse formula, but orthogonal uses transpose (A^(-1) = A^T) while involutory is its own inverse (A^(-1) = A).
- Idempotent vs Nilpotent: Idempotent "projects" (A^2 = A), nilpotent "annihilates" (A^k = O). Both can have det = 0, but for different structural reasons.
- Symmetric + Skew-symmetric: Every matrix = unique sum of symmetric and skew-symmetric parts. This decomposition is tested regularly.
- A skew-symmetric matrix of ODD order is always singular. Even order can be non-singular.