Orthogonal Matrix

An orthogonal matrix satisfies A^T A = I, meaning its columns form an orthonormal set, a structure important in JEE Advanced geometry and transformations. Taking determinants of A^T A = I gives det(A^T)·det(A) = det(I) = 1. Since det(A^T) = det(A), this becomes det(A)^2 = 1, so det(A) = +1 or det(A) = -1. Orthogonal matrices with determinant +1 represent rotations, while those with determinant -1 represent reflections or improper rotations. Option only +1 excludes the valid reflection case. Option 0 would make A singular, contradicting A^T A = I. Option any value with modulus less than 1 violates the derived constraint. Hence det(A) is +1 or -1. Plausibility check: the identity matrix is orthogonal with determinant +1 and a coordinate-swap reflection matrix is orthogonal with determinant -1, so both values occur, confirming the result det(A)^2 = 1. The characteristic polynomial encodes the trace as the negative coefficient of the second-highest power and the determinant as the constant term, tying these scalar invariants directly to the eigenvalue spectrum. For a two by two matrix this gives an immediate quadratic whose roots are the eigenvalues, and the same Vieta-style relations extend to larger matrices through the elementary symmetric functions of all eigenvalues.