$2\times2$ matrices are not big enough Olga Tausky-Todd had once said that

"If an assertion about matrices is false, there is usually a 2x2 matrix that reveals this."

There are, however, assertions about matrices that are true for $2\times2$ matrices but not for the larger ones. I came across one nice little example yesterday. Actually, every student who has studied first-year linear algebra should know that there are even assertions that are true for $3\times3$ matrices, but false for larger ones --- the rule of Sarrus is one obvious example; a question I answered last year provides another.
So, here is my question. What is your favourite assertion that is true for small matrices but not for larger ones? Here, $1\times1$ matrices are ignored because they form special cases too easily (otherwise, Tausky-Todd would have not made the above comment). The assertions are preferrably simple enough to understand, but their disproofs for larger matrices can be advanced or difficult.

Matrices of size $2 \times 2$ are fully understood in terms of the number of elementary multiplications required to multiply two matrices. This is not the case for even slightly bigger matrices. There is a gap between lower and upper bound for the number of multiplications required.
The product of two $2 \times 2$ matrices (denoted as case $<2,2,2>$) can be computed in $7$ elementary multiplications. Winograd showed that seven is actually the minimum number or multiplications required.
Bläser obtained a lower bound of $19$ multiplications for the product of $3 \times 3$ matrices. However, the best known algorithm for the non-commutative product needs $23$ multiplications. The solution of Laderman has not been improved since 1976. Recent work confirmed, that there are many other algorithms with $23$ multiplications.
The case $<2,2,3>$ (equivalent to $<2,3,2>$ and $<3,2,2>$) currently has a lower bound of $10$ and a known solution with $11$ multiplications. Hopcroft and Kerr published an algorithm for $<2,3,3>$ which requires $15$ multiplications, while the lower bound is $14$.
