Finding a matrix certificate given a non-negative polynomial I am trying to understand the following review question:

To show that the polynomial $$p(t)=t^3+4t^2-3t+3$$ is positive for all $t \ge 0$, we can use one of the following matrix certificates:


Initially, I thought it was sum of squares, but it won't match any of the answers. What does it mean by matrix certificate? Thank you for your help!
 A: From this handout, entitled Sum of Squares and authored by Sanjay Lall, it seems that in your context, a constant matrix $P$ may be called a matrix certificate for the positivity of $p(t)$ for $t\ge0$, if $P$ is positive definite and
$$
p(t)=\pmatrix{1&\sqrt{t}&\sqrt{t^2}&\sqrt{t^3}}
P\pmatrix{1\\ \sqrt{t}\\ \sqrt{t^2}\\ \sqrt{t^3}}.
$$
Apparently, it is called a "certificate" because its positive definiteness guarantees that $p(t)$ is a sum of squares (and hence positive).
If this interpretation is correct, then the answer to your question is $C$.
In general, since some cross terms are equal (e.g. $\sqrt{t}\sqrt{t^3}=\sqrt{t^2}\sqrt{t^2}$), the matrix representation of $p(t)$ is not unique. Therefore, even if $p$ is nonnegative, $P$ may fail to be positive definite. Thus one should consider a parametrised representation
$$
P(\lambda,\gamma,\mu)=P+\pmatrix{
0&0&-\lambda&-\gamma\\
0&2\lambda&\gamma&-\mu\\
-\lambda&\gamma&2\mu&0\\
-\gamma&-\mu&0&0}
$$
instead, and look for parameters $\lambda,\gamma$ and $\mu$ that makes $P(\lambda,\gamma,\mu)\succ0$. For instance, in your question both $B$ and $C=B(-2,0,0)$ are also representations of $p(t)$ --- and actually $B$ is a more natural representation than $C$ is --- but only $C$ can serve as a matrix certificate fro $p(t)>0$, because $B$ isn't positive definite.
