<s>[INST] Existence of some type matrix Is there square matrix $A$ of size $3$ with real entries such that 
$$
\operatorname{tr}(A)=0\text{ and }A^2+A^T=I.
$$ 
I have proved that there is not with size $2$ using definition of  "trace", but for size $3$ it becomes complicated.
Here is the sketch of the proof for $2$.
$$
a_{11}+a_{22}=0,\\
a_{11}^2+a_{12}a_{21}+a_{11}=1,\\
a_{11}a_{12}+a_{12}a_{22}+a_{21}=0,\\
a_{21}a_{11}+a_{21}a_{22}+a_{12}=0,\\
a_{12}a_{21}+a_{22}^2+a_{22}=1.
$$
Putting $a_{11}=-a_{22}$ it is easy to see that above inequalities can't be true together. 
Thanks!
 [/INST] The answer is no.
If we substitute $A^T=I-A^2$ into the transpose of $A^2+A^T=I$, namely $(A^T)^2+A=I$, we get $$(I-A^2)^2+A=I$$
Hence $I-2A^2+A^4+A=I$, or $$A^4-2A^2+A=0 ~~~~~~(\star)$$
must be satisfied by our matrix $A$.  This has just four roots: $0,1,\frac{-1+\sqrt{5}}{2},\frac{-1-\sqrt{5}}{2}$.  
Now, consider $B=QAQ^{-1}$, which is $A$ in Jordan canonical form.  By multiplying $(\star)$ on the left and right by $Q, Q^{-1}$ respectively, we must also have $$B^4-2B^2+B=0 ~~~~(\dagger)$$
$B$ has the same eigenvalues as $A$, which must sum to $0$ (counted by multiplicity) by the first condition.  Each block of $B$ must separately verify $(\dagger)$, hence the only possible nonzero $B$, up to reordering of the three blocks, is $$B=\left(\begin{smallmatrix}1&0&0\\0&\frac{-1+\sqrt{5}}{2}&0\\0&0&\frac{-1-\sqrt{5}}{2}\end{smallmatrix}\right)$$
Now, we plug $A=Q^{-1}BQ$ into $A^2+A^T=I$, to get $$Q^{-1}B^2Q+Q^TB^T(Q^{-1})^T=I$$
Multiplying on the left and right by $Q$ and $Q^{-1}$ respectively, and set $R=QQ^T$, to get
$$B^2+RBR^{-1}=I$$
It turns out that $$I-B^2=\left(\begin{smallmatrix}0&0&0\\0&\frac{-1+\sqrt{5}}{2}&0\\0&0&\frac{-1-\sqrt{5}}{2}\end{smallmatrix}\right)$$
But since $RBR^{-1}=(I-B^2)$, $B$ and $I-B^2$ must have the same eigenvalues.  They have two in common, but not all three -- $B$ has $1$ while $I-B^2$ has $0$.  Hence no such $A$ exists.
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