Project ForgeEngineering Mathematics
Linear Algebra
GATE 2017
Hard
Consider the matrix P = [[1/√2, 0, 1/√2], [0, 1, 0], [-1/√2, 0, 1/√2]]. Which one of the following statements about P is INCORRECT?
- ADeterminant of P is equal to 1.
- BP is orthogonal.
- CInverse of P is equal to its transpose.
- All eigenvalues of P are real numbers.
Explanation
P is orthogonal with det(P)=1, so P^-1 = P^T (statements A, B, C are all correct). Orthogonal matrices that are not symmetric — as this one is, representing a rotation — generally have complex-conjugate eigenvalue pairs alongside real ones; here P is a rotation about the y-axis, whose eigenvalues are 1 and e^(±iθ), not all real. So statement D is the incorrect one.
Source: GATE ME 2017 Session 1 (ME-1) (2017)
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