alleryStart learning →An ORTHOGONAL matrix $Q$ satisfies
A$Q^T Q = I$
B$\det Q = 1$ only
Call entries are 1 or -1
D$Q$ is diagonal
Answer & Solution
Correct answer: A. $Q^T Q = I$
1. ORTHOGONAL matrix: $Q^T Q = QQ^T = I$, i.e. $Q^{-1} = Q^T$.
2. Equivalently: the COLUMNS of $Q$ form an ORTHONORMAL set in $\mathbb{R}^n$ (and same for rows).
3. PROPERTY: orthogonal matrices preserve LENGTH ($\|Qx\| = \|x\|$) and ANGLES — they represent rotations and reflections.
4. $\det Q = \pm 1$ (rotation = +1, reflection = -1). So option B is too restrictive.
5. Options C, D describe special cases or wrong properties.
_Source: Sergei Treil, "Linear Algebra Done Wrong", §5.1 (Orthogonal matrices)._
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