<s>[INST] Criteria for $3 \times 3$ matrix to positive definite Here it is said that a $2\times 2$ matrix $A$ is positive definite if and only if $tr(A) >0$ and $det(A)>0$. This will not work if $A$ is $3\times 3$. But is there any way to enforce the positive definiteness of the matrix $A$ via the trace and determinant of $A$, if $A$ is of size $3\times 3$?
 [/INST] No, it is not possible. Take any two real numbers $a$ and $b$ such that $a^2+b^2=1$ and the matrix
$$
A=
\begin{pmatrix}
1 & a & b \\
a & a^2+1 & a b \\
b & a b & -a^2+2
\end{pmatrix}.
$$
Then $\operatorname{tr}(A)=4$, $\det(A)=1$, and $A$ is definite positive.
But if you take two real numbers $a$ and $b$ such that $a^2+b^2=5$ and you define
$$
B=
\begin{pmatrix}
1&a&b\\
a&a^2-1&ab\\
b&ab&-a^2+4
\end{pmatrix},
$$
then $\operatorname{tr}(B)=4$, $\det(B)=1$, but $B$ is not definite positive.
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