$2\times2$ 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.

Interesting although quite elementary is property for matrices made from consecutive integers numbers (or more generally from values of arithmetic progression) where rows make an arithmetic progression.
Only $2 \times 2$ matrices made from consecutive integers numbers are non-singular, matrices of higher dimension are singular.
For example  matrix 
$\begin{bmatrix}
1 & 2\\
3 & 4 \\
\end{bmatrix}$  is non-singular,
but matrices like $\begin{bmatrix}
 1 &  2 &  3 \\ 
 4 &  5 &  6   \\
 7  & 8 &  9   
   \end{bmatrix}$, $\begin{bmatrix}
 1 &   2 &  3 &  4 \\ 
 5 &  6 &  7 &  8 \\
 9  & 10 &  11 &  12 \\
 13 &  14  & 15 &  16 
\end{bmatrix}$,  $\dots$  are singular.
This can be applied for easy generating of singular matrices if needed.
