Every special matrix type has a "fingerprint" property that helps identify it quickly:

• Symmetric (A = $A^T$): All eigenvalues real. Can always be diagonalized.
• Skew-symmetric (A = -$A^T$): Diagonal = 0. Odd order => det = 0. Eigenvalues are 0 or purely imaginary.
• Orthogonal ($AA^T$ = I): Columns are orthonormal. det(A) = +/-1. Preserves distances.
• Idempotent ($A^2$ = A): Eigenvalues are only 0 or 1. trace(A) = rank(A).
• Involutory ($A^2$ = I): Self-inverse. Eigenvalues are +/-1.
• Nilpotent ($A^k$ = 0): All eigenvalues = 0. det = 0. trace = 0.

Quick test: If a JEE problem mentions any of these properties, immediately recall the fingerprint -- it usually eliminates 2-3 options instantly.