Given two symmetric and positive semidefinite square matrices A , B A, B , is it true that any matrix given as the product of m m copies of A A and n n copies of B B in a particular sequence must be dominated in the spectral norm by the ordered matrix product A m B n A^m B^n ? For example, is ‖ A A B A A B A B B ‖ ≤ ‖ A A A A A B B B B ‖ ? \begin{equation*} \| AABAABABB \| \leq \| AAAAABBBB \| ? \end{equation*} Drury [Electron J. Linear Algebra 18 (2009), pp. 13–20] has characterized precisely which disordered words have the property that an inequality of this type holds for all matrices A , B A,B . However, the 1 1 -parameter family of counterexamples Drury constructs for these characterizations is comprised of 3 × 3 3 \times 3 matrices, and thus as stated the characterization applies only for N × N N \times N matrices with N ≥ 3 N \geq 3 . In contrast, we prove that for 2 × 2 2 \times 2 matrices, the general rearrangement inequality holds for all disordered words. We also show that for larger N × N N \times N matrices, the general rearrangement inequality holds for all disordered words for most A , B A,B (in a sense of full measure) that are sufficiently small perturbations of the identity.