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!
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.