Rank and nullity of a rotation

Let RR be the 3×33\times 3 matrix of a rotation in R3\mathbb{R}^3 (a composition of rotations about the xx-, yy-, and zz-axes by fixed angles). Compute (rankR)2+(nullityR)2(\operatorname{rank} R)^2 + (\operatorname{nullity} R)^2.

Show hints (2)+
  1. Is a rotation invertible? Consider detR\det R and whether it changes lengths.
  2. It's orthogonal with det=1\det=1, so rank 33, nullity 00: 9+09+0.

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9

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Asked at: Jane Street, Two Sigma

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