TITLE: $2\times2$ matrices are not big enough
QUESTION [41 upvotes]: 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.

REPLY [2 votes]: Let  $A \in M_2(\mathbb R)$, $A$ is invertible or $A$ is 
 nilpotent or $A$ is diagonalizable
Proof: Let $\chi_A$ be its characteristic polynomial, then $\chi_A = X^2-$ tr$(A)X + \det A$.
If $\det A \neq 0$ then $A$ is invertible.
Else if tr$(A) = 0$, then by  Cayley-Hamilton theorem, $A^2 = 0$ i.e. $A$ is nilpotent.
Else $X^2-$ tr$(A)X = X(X- $tr $(A))$ has distinct simple roots and vanishes $A$, hence $A$ is diagonalizable.