$2\times2$ 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.

Any two rotation matrices commute.

I like this one: two matrices are similar (conjugate) if and only if they have the same minimal and characteristic polynomials and the same dimensions of eigenspaces corresponding to the same eigenvalue. This statement is true for all $n\times n$ matrices with $n\leq6$, but is false for $n\geq7$.

A matrix is called doubly stochastic if all its entries are nonnegative and each row and column sums to 1. A matrix is called orthostochastic if it is the entry-wise square of some orthogonal matrix. In general, it is easy to see that every orthostochastic matrix is doubly stochastic.

For $2\times 2$ matrices, every doubly stochastic matrix is orthostochastic, but for $3\times 3$ matrices (and hence anything larger) we have the counterexample $$ \frac{1}{2}\left(\begin{matrix} 1 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 1 \end{matrix}\right)$$